diff --git a/docs/src/manual/implementing.md b/docs/src/manual/implementing.md
index cf63fc77d4..13809721d4 100644
--- a/docs/src/manual/implementing.md
+++ b/docs/src/manual/implementing.md
@@ -453,25 +453,23 @@ MOI.supports_add_constrained_variables(::Optimizer, ::Type{Reals}) = false
 ### Incremental and `copy_to`
 
 If you implement the incremental interface, you have the option of also
-implementing [`copy_to`](@ref). If you don't want to implement
-[`copy_to`](@ref), e.g., because the solver has no API for building the problem
-in a single function call, define the following fallback:
+implementing [`copy_to`](@ref).
+
+If you don't want to implement [`copy_to`](@ref), e.g., because the solver has
+no API for building the problem in a single function call, define the following
+fallback:
 ```julia
+# If you support VariableName and ConstraintName...
+MOI.supports_incremental_interface(::Optimizer, copy_names::Bool) = true
+# Otherwise...
+MOI.supports_incremental_interface(::Optimizer, copy_names::Bool) = !copy_names
+
 function MOI.copy_to(dest::Optimizer, src::MOI.ModelLike; kwargs...)
     return MOI.Utilities.automatic_copy_to(dest, src; kwargs...)
 end
-
-function MOI.Utilities.supports_default_copy_to(
-    model::Optimizer,
-    copy_names::Bool,
-)
-    # If you support names...
-    return true
-    # Otherwise...
-    return !copy_names
-end
 ```
-See [`Utilities.supports_default_copy_to`](@ref) for more details.
+See [`supports_incremental_interface`](@ref) for more details on whether to
+implement the `true` or `!copy_names` version.
 
 ## [Names](@id implement_names)
 
diff --git a/docs/src/reference/models.md b/docs/src/reference/models.md
index f9a736c3bb..177ee06cb5 100644
--- a/docs/src/reference/models.md
+++ b/docs/src/reference/models.md
@@ -28,6 +28,7 @@ is_empty
 empty!
 write_to_file
 read_from_file
+supports_incremental_interface
 copy_to
 ```
 
diff --git a/docs/src/submodules/Test/overview.md b/docs/src/submodules/Test/overview.md
index f6a48fdd5a..545abb1f0b 100644
--- a/docs/src/submodules/Test/overview.md
+++ b/docs/src/submodules/Test/overview.md
@@ -22,7 +22,7 @@ so that all solvers can benefit.
 The skeleton below can be used for the wrapper test file of a solver named
 `FooBar`. Remove unnecessary tests as appropriate, for example tests for
 features that the solver does not support (tests are not skipped depending
-on the value of `supports`). 
+on the value of `supports`).
 
 ```julia
 # ============================ /test/MOI_wrapper.jl ============================
@@ -61,10 +61,10 @@ function test_SolverName()
     @test MOI.get(OPTIMIZER, MOI.SolverName()) == "FooBar"
 end
 
-function test_supports_default_copy_to()
-    @test MOI.Utilities.supports_default_copy_to(OPTIMIZER, false)
+function test_supports_incremental_interface()
+    @test MOI.supports_incremental_interface(OPTIMIZER, false)
     # Use `@test !...` if names are not supported
-    @test MOI.Utilities.supports_default_copy_to(OPTIMIZER, true)
+    @test MOI.supports_incremental_interface(OPTIMIZER, true)
 end
 
 function test_unittest()
diff --git a/docs/src/submodules/Utilities/reference.md b/docs/src/submodules/Utilities/reference.md
index 36a21cb558..e40184e8a4 100644
--- a/docs/src/submodules/Utilities/reference.md
+++ b/docs/src/submodules/Utilities/reference.md
@@ -58,7 +58,6 @@ Utilities.latex_formulation
 ```@docs
 Utilities.automatic_copy_to
 Utilities.default_copy_to
-Utilities.supports_default_copy_to
 ```
 
 ### [Allocate-Load API](@id allocate_load_api_ref)
diff --git a/src/Bridges/bridge_optimizer.jl b/src/Bridges/bridge_optimizer.jl
index 1684a7b0af..417c3a2b96 100644
--- a/src/Bridges/bridge_optimizer.jl
+++ b/src/Bridges/bridge_optimizer.jl
@@ -358,11 +358,11 @@ end
 function MOI.copy_to(mock::AbstractBridgeOptimizer, src::MOI.ModelLike; kws...)
     return MOIU.automatic_copy_to(mock, src; kws...)
 end
-function MOIU.supports_default_copy_to(
+function MOI.supports_incremental_interface(
     b::AbstractBridgeOptimizer,
     copy_names::Bool,
 )
-    return MOIU.supports_default_copy_to(b.model, copy_names)
+    return MOI.supports_incremental_interface(b.model, copy_names)
 end
 
 # References
diff --git a/src/MathOptInterface.jl b/src/MathOptInterface.jl
index 5a87848f3f..45376fc7e8 100644
--- a/src/MathOptInterface.jl
+++ b/src/MathOptInterface.jl
@@ -94,6 +94,35 @@ Empty the model, that is, remove all variables, constraints and model attributes
 """
 function empty! end
 
+"""
+    supports_incremental_interface(model::ModelLike, copy_names::Bool)
+
+Return a `Bool` indicating whether `model` supports building incrementally via
+[`add_variable`](@ref) and [`add_constraint`](@ref).
+
+`copy_names` is a `Bool` indicating whether the user wishes to set
+[`VariableName`](@ref) and [`ConstraintName`](@ref) attributes.
+If `model` supports the incremental interface but does not support name
+attributes, define
+```julia
+supports_incremental_interface(::MyNewModel, copy_names::Bool) = !copy_names
+```
+
+The main purpose of this function is to determine whether a model can be loaded
+into `model` incrementally or whether it should be cached and copied at once
+instead.
+
+This is used in two places to determine whether to add a cache:
+1. A first cache can be used to store the model as entered by the user as well
+   as the names of variables and constraints. This cache is created if this
+   function returns `false` when `copy_names` is `true`.
+2. If bridges are used, then a second cache can be used to store the bridged
+   model with unnamed variables and constraints. This cache is created if this
+   function returns `false` when `copy_names` is `false`.
+```
+"""
+supports_incremental_interface(::ModelLike, ::Bool) = false
+
 """
     copy_to(dest::ModelLike, src::ModelLike; copy_names=true, warn_attributes=true)
 
diff --git a/src/Test/contconic.jl b/src/Test/contconic.jl
index e943734ce9..60e77cadea 100644
--- a/src/Test/contconic.jl
+++ b/src/Test/contconic.jl
@@ -11,7 +11,7 @@ function _lin1test(model::MOI.ModelLike, config::TestConfig, vecofvars::Bool)
     #       x>=0 y>=0 z>=0
     # Opt obj = -11, soln x = 1, y = 0, z = 2
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -157,7 +157,7 @@ function _lin2test(model::MOI.ModelLike, config::TestConfig, vecofvars::Bool)
     # Opt solution = -82
     # x = -4, y = -3, z = 16, s == 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -346,7 +346,7 @@ function lin3test(model::MOI.ModelLike, config::TestConfig)
     # s.t. -1 + x ∈ R₊
     #       1 + x ∈ R₋
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -436,7 +436,7 @@ function lin4test(model::MOI.ModelLike, config::TestConfig)
     # s.t. -1 + x ∈ R₊
     #           x ∈ R₋
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -524,7 +524,7 @@ function _norminf1test(
     #      y == 1/2
     #      x >= ||(y,z)||_∞
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -671,7 +671,7 @@ function norminf2test(model::MOI.ModelLike, config::TestConfig)
     #      -1 + x ∈ R₋
     #       (x,y) ∈ NormInf₂
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -772,7 +772,7 @@ function norminf3test(model::MOI.ModelLike, config::TestConfig)
     #      (1 .+ y) in Nonnegatives(n)
     # let n = 3. optimal solution: y .= -1, x = 2
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(
@@ -884,7 +884,7 @@ function _normone1test(
     #      y == 1/2
     #      x >= ||(y,z)||_1
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -1027,7 +1027,7 @@ function normone2test(model::MOI.ModelLike, config::TestConfig)
     #      -1 + x ∈ R₋
     #       (x,y) ∈ NormOne₂
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -1128,7 +1128,7 @@ function normone3test(model::MOI.ModelLike, config::TestConfig)
     #      (1 .+ y) in Nonnegatives(n)
     # let n = 3. optimal solution: y .= -1, x = 4
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(
@@ -1234,7 +1234,7 @@ function _soc1test(model::MOI.ModelLike, config::TestConfig, vecofvars::Bool)
     #  st  x == 1
     #      x >= ||(y,z)||
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -1365,7 +1365,7 @@ function _soc2test(model::MOI.ModelLike, config::TestConfig, nonneg::Bool)
     #        1 - t ∈ {0}
     #      (t,x,y) ∈ SOC₃
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -1530,7 +1530,7 @@ function soc3test(model::MOI.ModelLike, config::TestConfig)
     #      -1 + x ∈ R₋
     #       (x,y) ∈ SOC₂
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -1638,7 +1638,7 @@ function soc4test(model::MOI.ModelLike, config::TestConfig)
     # Like SOCINT1 but with copies of variables and integrality relaxed
     # Tests out-of-order indices in cones
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -1760,7 +1760,7 @@ function _rotatedsoc1test(
     #     [0.0] - [-y    ] SOCRotated
     #     [0.0] - [    -z] SOCRotated
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -1913,7 +1913,7 @@ function rotatedsoc2test(model::MOI.ModelLike, config::TestConfig)
     b = [-2, -1, 1 / 2]
     c = [0.0, 0.0, 0.0]
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -2036,7 +2036,7 @@ function rotatedsoc3test(
     # [t1/√2, t2/√2, x] in SOC4
     # [x1/√2, u/√2,  v] in SOC3
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -2232,7 +2232,7 @@ function rotatedsoc4test(
     # We conclude that the optimal solution is x = y = t = u = 1
     # with objective value 2.
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -2335,7 +2335,7 @@ function _geomean1test(
     # Therefore xyz ≤ 1
     # This can be attained using x = y = z = 1 so it is optimal.
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -2451,7 +2451,7 @@ function _geomean2test(model::MOI.ModelLike, config::TestConfig, vecofvars)
     #     x_1 == x_2, ..., x_9 == 1
     # the optimal solution is 1 with optimal value 1
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -2575,7 +2575,7 @@ function _geomean3test(model::MOI.ModelLike, config::TestConfig, vecofvars)
     #     x <= 2
     # the optimal solution is (t, x) = (2, 2) with objective value 4
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -2696,7 +2696,7 @@ function _exp1test(model::MOI.ModelLike, config::TestConfig, vecofvars::Bool)
     #      x == 1
     #      y == 2
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -2813,7 +2813,7 @@ function exp2test(model::MOI.ModelLike, config::TestConfig)
     atol = config.atol
     rtol = config.rtol
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -2991,7 +2991,7 @@ function exp3test(model::MOI.ModelLike, config::TestConfig)
     atol = config.atol
     rtol = config.rtol
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -3109,7 +3109,7 @@ function _dualexp1test(
     # w == 1
     # (u, v, w) ∈ DualExponentialCone
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -3257,7 +3257,7 @@ function _pow1test(model::MOI.ModelLike, config::TestConfig, vecofvars::Bool)
     #     v + β = 0
     #     w = -1
     a = 0.9
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -3397,7 +3397,7 @@ function _dualpow1test(
     # hence you get v = 0.1 and u = 0.9.
     # The same works for other values of exponent as key word argument
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -3538,7 +3538,7 @@ function relentr1test(model::MOI.ModelLike, config::TestConfig)
     # Optimal solution is:
     # u = 2*log(2/1) + 3*log(3/5) ≈ -0.1461825
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(
@@ -3615,7 +3615,7 @@ function normspec1test(model::MOI.ModelLike, config::TestConfig)
     # Singular values are [sqrt(3), sqrt(2)], so optimal solution is:
     # t = sqrt(3)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(
@@ -3689,7 +3689,7 @@ function normspec2test(model::MOI.ModelLike, config::TestConfig)
     # Singular values are [sqrt(3), sqrt(2)], so optimal solution is:
     # t = sqrt(3)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(
@@ -3767,7 +3767,7 @@ function normnuc1test(model::MOI.ModelLike, config::TestConfig)
     # Singular values are [sqrt(3), sqrt(2)], so optimal solution is:
     # t = sqrt(3) + sqrt(2)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(
@@ -3844,7 +3844,7 @@ function normnuc2test(model::MOI.ModelLike, config::TestConfig)
     # Singular values are [sqrt(3), sqrt(2)], so optimal solution is:
     # t = sqrt(3) + sqrt(2)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(
@@ -3934,7 +3934,7 @@ function _psd0test(
     # X = ⎜       ⎟           y = 2
     #     ⎝ 1   1 ⎠
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -4130,7 +4130,7 @@ function _psd1test(
     α = √(3 - 2obj - 4x2) / 2
     β = k * α
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -4308,7 +4308,7 @@ function psdt2test(model::MOI.ModelLike, config::TestConfig)
     rtol = config.rtol
     # Caused getdual to fail on SCS and Mosek
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -4494,7 +4494,7 @@ function _psd3test(
     atol = config.atol
     rtol = config.rtol
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(model, MOI.VectorAffineFunction{T}, psdcone)
@@ -4631,7 +4631,7 @@ function _det1test(
     #           |_________|
     #            -Q22 ≥ -1
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -4808,7 +4808,7 @@ function _det2test(model::MOI.ModelLike, config::TestConfig, detcone)
     mat = Float64[3 2 1; 2 2 1; 1 1 3]
     matL = Float64[3, 2, 2, 1, 1, 3]
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(model, MOI.ObjectiveFunction{MOI.SingleVariable}())
     @test MOI.supports(model, MOI.ObjectiveSense())
     @test MOI.supports_constraint(
diff --git a/src/Test/contlinear.jl b/src/Test/contlinear.jl
index ac9aadd64e..a5e3284f3a 100644
--- a/src/Test/contlinear.jl
+++ b/src/Test/contlinear.jl
@@ -9,7 +9,7 @@ function linear1test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     # st   x + y <= 1   (x + y - 1 ∈ Nonpositives)
     #       x, y >= 0   (x, y ∈ Nonnegatives)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -571,7 +571,7 @@ function linear2test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     # s.t. x + y <= 1
     # x, y >= 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -681,7 +681,7 @@ function linear3test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     # s.t. x >= 0
     #      x >= 3
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -820,7 +820,7 @@ function linear4test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     atol = config.atol
     rtol = config.rtol
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -930,7 +930,7 @@ function linear5test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     #
     #   solution: x = 1.3333333, y = 1.3333333, objv = 2.66666666
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -1099,7 +1099,7 @@ function linear6test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     atol = config.atol
     rtol = config.rtol
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -1238,7 +1238,7 @@ function linear7test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     #                  w ==   1     (c2)
     # i.e. z == w == 1
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -1389,7 +1389,7 @@ function linear8atest(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     # s.t. 2x+y <= -1
     # x,y >= 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -1476,7 +1476,7 @@ function linear8btest(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     # s.t. -x+2y <= 0
     # x,y >= 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -1553,7 +1553,7 @@ function linear8ctest(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     # s.t. x-y == 0
     # x,y >= 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -1639,7 +1639,7 @@ function linear9test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     #   solution: (59.0909, 36.3636)
     #   objv: 71818.1818
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -1746,7 +1746,7 @@ function linear10test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     #       s.t.  5 <= x + y <= 10
     #                  x,  y >= 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -1952,7 +1952,7 @@ function linear10btest(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     #       s.t.  -1 <= x + y <= 10
     #                   x,  y >= 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -2071,7 +2071,7 @@ function linear11test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     #      w >= 0, z <= 0
     # sol: w = 1, z = 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -2159,7 +2159,7 @@ function linear12test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     #      y <= 2
     # x,y >= 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -2250,7 +2250,7 @@ function linear13test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     # s.t. 2x + 3y >= 1
     #      x - y == 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports_constraint(
         model,
         MOI.ScalarAffineFunction{T},
@@ -2329,7 +2329,7 @@ function linear14test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     #      x, y, z >= 0
     #      z <= 1
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
@@ -2492,7 +2492,7 @@ function linear15test(model::MOI.ModelLike, config::TestConfig{T}) where {T}
     # minimize 0
     # s.t. 0 == 0
     #      x == 1
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
diff --git a/src/Test/contquadratic.jl b/src/Test/contquadratic.jl
index 3c44de18fa..55ee0e7f1e 100644
--- a/src/Test/contquadratic.jl
+++ b/src/Test/contquadratic.jl
@@ -9,7 +9,7 @@ function qp1test(model::MOI.ModelLike, config::TestConfig)
     #     x +  y      >= 1 (c2)
     #     x, y, z \in R
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}(),
@@ -120,7 +120,7 @@ function qp2test(model::MOI.ModelLike, config::TestConfig)
     #     x +  y      >= 1 (c2)
     #     x, y, z \in R
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}(),
@@ -274,7 +274,7 @@ function qp3test(model::MOI.ModelLike, config::TestConfig)
     #       s.t.  x, y >= 0
     #             x + y = 1
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarQuadraticFunction{Float64}}(),
@@ -414,7 +414,7 @@ function qcp1test(model::MOI.ModelLike, config::TestConfig)
     # Optimal solution
     # x = 1/2, y = 7/4
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -540,7 +540,7 @@ function qcp2test(model::MOI.ModelLike, config::TestConfig)
     # Max x
     # s.t. x^2 <= 2 (c)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -618,7 +618,7 @@ function qcp3test(model::MOI.ModelLike, config::TestConfig)
     # Min -x
     # s.t. x^2 <= 2
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -701,7 +701,7 @@ function _qcp4test(model::MOI.ModelLike, config::TestConfig, less_than::Bool)
     # s.t. x^2 + x * y + y^2 <= 3 (c)
     #      y == 1
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -818,7 +818,7 @@ function ncqcp1test(model::MOI.ModelLike, config::TestConfig)
     # s.t. x * y <= 4 (c)
     #      x, y >= 1
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -903,7 +903,7 @@ function ncqcp2test(model::MOI.ModelLike, config::TestConfig)
     #      x * x == 4 (c2)
     #      x, y >= 0
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports_constraint(
         model,
         MOI.ScalarQuadraticFunction{Float64},
@@ -990,7 +990,7 @@ function socp1test(model::MOI.ModelLike, config::TestConfig)
     #      x^2 + y^2 <= t^2 (c2)
     #      t >= 0 (bound)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
diff --git a/src/Test/intconic.jl b/src/Test/intconic.jl
index f698129ad1..15d925800d 100644
--- a/src/Test/intconic.jl
+++ b/src/Test/intconic.jl
@@ -10,7 +10,7 @@ function intsoc1test(model::MOI.ModelLike, config::TestConfig)
     #      x >= ||(y,z)||
     #      (y,z) binary
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
diff --git a/src/Test/intlinear.jl b/src/Test/intlinear.jl
index 8a245b8e3b..c2cb8020e0 100644
--- a/src/Test/intlinear.jl
+++ b/src/Test/intlinear.jl
@@ -13,7 +13,7 @@ function int1test(model::MOI.ModelLike, config::TestConfig)
     #         y is integer: 0 <= y <= 10
     #         z is binary
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -147,7 +147,7 @@ function int2test(model::MOI.ModelLike, config::TestConfig)
     atol = config.atol
     rtol = config.rtol
     @testset "SOSI" begin
-        @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+        @test MOI.supports_incremental_interface(model, false) #=copy_names=#
         @test MOI.supports(
             model,
             MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -266,7 +266,7 @@ function int2test(model::MOI.ModelLike, config::TestConfig)
         end
     end
     @testset "SOSII" begin
-        @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+        @test MOI.supports_incremental_interface(model, false) #=copy_names=#
         @test MOI.supports(
             model,
             MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -426,7 +426,7 @@ function int3test(model::MOI.ModelLike, config::TestConfig)
     MOI.empty!(model)
     @test MOI.is_empty(model)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -519,7 +519,7 @@ function knapsacktest(model::MOI.ModelLike, config::TestConfig)
     MOI.empty!(model)
     @test MOI.is_empty(model)
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{Float64}}(),
@@ -1032,7 +1032,7 @@ function _semitest(
     atol = config.atol
     rtol = config.rtol
 
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     @test MOI.supports(
         model,
         MOI.ObjectiveFunction{MOI.ScalarAffineFunction{T}}(),
diff --git a/src/Test/modellike.jl b/src/Test/modellike.jl
index 265254eb05..da3d11ec6e 100644
--- a/src/Test/modellike.jl
+++ b/src/Test/modellike.jl
@@ -79,7 +79,7 @@ function nametest(model::MOI.ModelLike)
 
     @testset "Name test with $(typeof(model))" begin
         MOI.empty!(model)
-        @test MOIU.supports_default_copy_to(model, true) #=copy_names=#
+        @test MOI.supports_incremental_interface(model, true) #=copy_names=#
         @test MOI.supports(model, MOI.Name())
         @test !(MOI.Name() in MOI.get(model, MOI.ListOfModelAttributesSet()))
         @test MOI.get(model, MOI.Name()) == ""
@@ -352,7 +352,7 @@ end
 # Taken from https://github.com/jump-dev/MathOptInterfaceUtilities.jl/issues/41
 function validtest(model::MOI.ModelLike)
     MOI.empty!(model)
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     v = MOI.add_variables(model, 2)
     @test MOI.is_valid(model, v[1])
     @test MOI.is_valid(model, v[2])
@@ -404,7 +404,7 @@ end
 
 function emptytest(model::MOI.ModelLike)
     MOI.empty!(model)
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     # Taken from LIN1
     v = MOI.add_variables(model, 3)
     @test MOI.supports_constraint(
@@ -700,7 +700,7 @@ end
 
 function start_values_test(dest::MOI.ModelLike, src::MOI.ModelLike)
     MOI.empty!(dest)
-    @test MOIU.supports_default_copy_to(src, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(src, false) #=copy_names=#
     x, y, z = MOI.add_variables(src, 3)
     fz = MOI.SingleVariable(z)
     a = MOI.add_constraint(src, x, MOI.EqualTo(1.0))
@@ -784,7 +784,7 @@ function start_values_test(dest::MOI.ModelLike, src::MOI.ModelLike)
 end
 
 function copytest(dest::MOI.ModelLike, src::MOI.ModelLike; copy_names = false)
-    @test MOIU.supports_default_copy_to(src, true) #=copy_names=#
+    @test MOI.supports_incremental_interface(src, true) #=copy_names=#
     MOI.empty!(src)
     MOI.empty!(dest)
     MOI.set(src, MOI.Name(), "ModelName")
@@ -1000,7 +1000,7 @@ Test whether the model returns ListOfVariableIndices and ListOfConstraintIndices
 sorted by creation time.
 """
 function orderedindicestest(model::MOI.ModelLike)
-    @test MOIU.supports_default_copy_to(model, false) #=copy_names=#
+    @test MOI.supports_incremental_interface(model, false) #=copy_names=#
     MOI.empty!(model)
     v1 = MOI.add_variable(model)
     @test MOI.get(model, MOI.ListOfVariableIndices()) == [v1]
diff --git a/src/Utilities/cachingoptimizer.jl b/src/Utilities/cachingoptimizer.jl
index 563aef2f7c..127f5017d7 100644
--- a/src/Utilities/cachingoptimizer.jl
+++ b/src/Utilities/cachingoptimizer.jl
@@ -239,8 +239,11 @@ function MOI.copy_to(m::CachingOptimizer, src::MOI.ModelLike; kws...)
     return MOI.copy_to(m.model_cache, src; kws...)
 end
 
-function supports_default_copy_to(model::CachingOptimizer, copy_names::Bool)
-    return supports_default_copy_to(model.model_cache, copy_names)
+function MOI.supports_incremental_interface(
+    model::CachingOptimizer,
+    copy_names::Bool,
+)
+    return MOI.supports_incremental_interface(model.model_cache, copy_names)
 end
 
 function MOI.empty!(m::CachingOptimizer)
diff --git a/src/Utilities/copy.jl b/src/Utilities/copy.jl
index cff63ceb3f..d3c3a3826b 100644
--- a/src/Utilities/copy.jl
+++ b/src/Utilities/copy.jl
@@ -5,7 +5,7 @@
                       copy_names::Bool=true,
                       filter_constraints::Union{Nothing, Function}=nothing)
 
-Use [`Utilities.supports_default_copy_to`](@ref) and
+Use [`MathOptInterface.supports_incremental_interface`](@ref) and
 [`Utilities.supports_allocate_load`](@ref) to automatically choose between
 [`Utilities.default_copy_to`](@ref) or [`Utilities.allocate_load`](@ref) to
 apply the copy operation.
@@ -20,7 +20,7 @@ function automatic_copy_to(
     copy_names::Bool = true,
     filter_constraints::Union{Nothing,Function} = nothing,
 )
-    if supports_default_copy_to(dest, copy_names)
+    if MOI.supports_incremental_interface(dest, copy_names)
         default_copy_to(dest, src, copy_names, filter_constraints)
     elseif supports_allocate_load(dest, copy_names)
         allocate_load(dest, src, copy_names, filter_constraints)
@@ -33,47 +33,7 @@ function automatic_copy_to(
     end
 end
 
-"""
-    supports_default_copy_to(model::ModelLike, copy_names::Bool)
-
-Return a `Bool` indicating whether the model `model` supports
-[`default_copy_to(model, src, copy_names=copy_names)`](@ref) if all the
-attributes set to `src` and constraints added to `src` are supported by `model`.
-
-This function can be used to determine whether a model can be loaded into
-`model` incrementally or whether it should be cached and copied at once instead.
-This is used by JuMP to determine whether to add a cache or not in two
-situations:
-1. A first cache can be used to store the model as entered by the user as well
-   as the names of variables and constraints. This cache is created if this
-   function returns `false` when `copy_names` is `true`.
-2. If bridges are used, then a second cache can be used to store the bridged
-   model with unnamed variables and constraints. This cache is created if this
-   function returns `false` when `copy_names` is `false`.
-
-## Examples
-
-If [`MathOptInterface.set`](@ref), [`MathOptInterface.add_variable`](@ref) and
-[`MathOptInterface.add_constraint`](@ref) are implemented for a model of type
-`MyModel` and names are supported, then [`MathOptInterface.copy_to`](@ref) can
-be implemented as
-```julia
-MOI.Utilities.supports_default_copy_to(model::MyModel, copy_names::Bool) = true
-function MOI.copy_to(dest::MyModel, src::MOI.ModelLike; kws...)
-    return MOI.Utilities.automatic_copy_to(dest, src; kws...)
-end
-```
-The [`Utilities.automatic_copy_to`](@ref) function automatically redirects to
-[`Utilities.default_copy_to`](@ref).
-
-If names are not supported, simply change the first line by
-```julia
-MOI.supports_default_copy_to(model::MyModel, copy_names::Bool) = !copy_names
-```
-The [`Utilities.default_copy_to`](@ref) function automatically throws an helpful
-error in case `copy_to` is called with `copy_names` equal to `true`.
-"""
-supports_default_copy_to(model::MOI.ModelLike, copy_names::Bool) = false
+@deprecate supports_default_copy_to MOI.supports_incremental_interface
 
 """
     _index_to_variable(i::Int)
@@ -654,8 +614,8 @@ end
 
 Implements `MOI.copy_to(dest, src)` by adding the variables and then the
 constraints and attributes incrementally. The function
-[`supports_default_copy_to`](@ref) can be used to check whether `dest` supports
-the copying a model incrementally.
+[`MathOptInterface.supports_incremental_interface`](@ref) can be used to check
+whether `dest` supports the copying a model incrementally.
 
 If the `filter_constraints` arguments is given, only the constraints for which
 this function returns `true` will be copied. This function is given a
diff --git a/src/Utilities/mockoptimizer.jl b/src/Utilities/mockoptimizer.jl
index 1e393fb4ba..e4e35af863 100644
--- a/src/Utilities/mockoptimizer.jl
+++ b/src/Utilities/mockoptimizer.jl
@@ -754,9 +754,12 @@ end
 function MOI.copy_to(mock::MockOptimizer, src::MOI.ModelLike; kws...)
     return automatic_copy_to(mock, src; kws...)
 end
-function supports_default_copy_to(mock::MockOptimizer, copy_names::Bool)
+function MOI.supports_incremental_interface(
+    mock::MockOptimizer,
+    copy_names::Bool,
+)
     return !mock.needs_allocate_load &&
-           supports_default_copy_to(mock.inner_model, copy_names)
+           MOI.supports_incremental_interface(mock.inner_model, copy_names)
 end
 
 # Allocate-Load Interface
diff --git a/src/Utilities/model.jl b/src/Utilities/model.jl
index 46596e4eec..81de8d35f9 100644
--- a/src/Utilities/model.jl
+++ b/src/Utilities/model.jl
@@ -772,7 +772,7 @@ end
 function MOI.copy_to(dest::AbstractModel, src::MOI.ModelLike; kws...)
     return automatic_copy_to(dest, src; kws...)
 end
-supports_default_copy_to(model::AbstractModel, copy_names::Bool) = true
+MOI.supports_incremental_interface(::AbstractModel, ::Bool) = true
 
 # Allocate-Load Interface
 # Even if the model does not need it and use default_copy_to, it could be used
diff --git a/src/Utilities/universalfallback.jl b/src/Utilities/universalfallback.jl
index eb7d26b96c..d12547f29c 100644
--- a/src/Utilities/universalfallback.jl
+++ b/src/Utilities/universalfallback.jl
@@ -125,8 +125,11 @@ function MOI.copy_to(uf::UniversalFallback, src::MOI.ModelLike; kws...)
     return MOIU.automatic_copy_to(uf, src; kws...)
 end
 
-function supports_default_copy_to(uf::UniversalFallback, copy_names::Bool)
-    return supports_default_copy_to(uf.model, copy_names)
+function MOI.supports_incremental_interface(
+    uf::UniversalFallback,
+    copy_names::Bool,
+)
+    return MOI.supports_incremental_interface(uf.model, copy_names)
 end
 
 # References
diff --git a/src/instantiate.jl b/src/instantiate.jl
index 11bd9dd2b8..9ec1ce1b9a 100644
--- a/src/instantiate.jl
+++ b/src/instantiate.jl
@@ -91,26 +91,26 @@ function _instantiate_and_check(optimizer_constructor::OptimizerWithAttributes)
 end
 
 """
-    instantiate(optimizer_constructor,
-                with_bridge_type::Union{Nothing, Type}=nothing,
-                with_names::Bool=false)
+    instantiate(
+        optimizer_constructor,
+        with_bridge_type::Union{Nothing, Type} = nothing,
+        with_names::Bool = false,
+    )
 
-Creates an instance of optimizer either by calling
-`optimizer_constructor.optimizer_constructor()` and setting the parameters in
-`optimizer_constructor.params` if `optimizer_constructor` is a
-[`OptimizerWithAttributes`](@ref) or by calling `optimizer_constructor()`
-if `optimizer_constructor` is callable.
+Creates an instance of optimizer by either:
+ * calling `optimizer_constructor.optimizer_constructor()` and setting the
+   parameters in `optimizer_constructor.params` if `optimizer_constructor` is a
+   [`OptimizerWithAttributes`](@ref)
+*  calling `optimizer_constructor()` if `optimizer_constructor` is callable.
 
 If `with_bridge_type` is not `nothing`, it enables all the bridges defined in
-the MathOptInterface.Bridges submodule with coefficient type
-`with_bridge_type`.
+the MathOptInterface.Bridges submodule with coefficient type `with_bridge_type`.
 
 If the optimizer created by `optimizer_constructor` does not support loading the
-problem incrementally or does not support names and `with_names` is `true` (see
-[`Utilities.supports_default_copy_to`](@ref)) then a
+problem incrementally (see [`supports_incremental_interface`](@ref)), or does
+not support names and `with_names` is `true`, then a
 [`Utilities.CachingOptimizer`](@ref) is added to store a cache of the bridged
-model.
-Hence set `with_names` to `true` if names might be set.
+model. Hence set `with_names` to `true` if names might be set.
 """
 function instantiate(
     optimizer_constructor;
@@ -121,7 +121,7 @@ function instantiate(
     if with_bridge_type === nothing
         return optimizer
     end
-    if !Utilities.supports_default_copy_to(optimizer, with_names)
+    if !supports_incremental_interface(optimizer, with_names)
         universal_fallback =
             Utilities.UniversalFallback(Utilities.Model{with_bridge_type}())
         optimizer = Utilities.CachingOptimizer(universal_fallback, optimizer)
diff --git a/test/Utilities/cachingoptimizer.jl b/test/Utilities/cachingoptimizer.jl
index 79336df7dd..8315e2be1c 100644
--- a/test/Utilities/cachingoptimizer.jl
+++ b/test/Utilities/cachingoptimizer.jl
@@ -610,7 +610,7 @@ function MOI.add_constrained_variables(
     return MOI.add_constrained_variables(model.inner, set)
 end
 
-MOI.Utilities.supports_default_copy_to(::NoFreeVariables, names::Bool) = !names
+MOI.supports_incremental_interface(::NoFreeVariables, names::Bool) = !names
 function MOI.copy_to(dest::NoFreeVariables, src::MOI.ModelLike; kwargs...)
     return MOI.Utilities.automatic_copy_to(dest, src; kwargs...)
 end
diff --git a/test/Utilities/copy.jl b/test/Utilities/copy.jl
index 4e45549330..3014a6868b 100644
--- a/test/Utilities/copy.jl
+++ b/test/Utilities/copy.jl
@@ -38,8 +38,8 @@ end
     end
 end
 @testset "Default" begin
-    @test !MOIU.supports_default_copy_to(DummyModel(), false)
-    @test !MOIU.supports_default_copy_to(DummyModel(), true)
+    @test !MOI.supports_incremental_interface(DummyModel(), false)
+    @test !MOI.supports_incremental_interface(DummyModel(), true)
     model = MOIU.Model{Float64}()
     MOIT.failcopytestc(model)
     MOIT.failcopytestia(model)
@@ -96,7 +96,10 @@ function MOI.empty!(model::ConstrainedVariablesModel)
     return empty!(model.added_constrained)
 end
 
-function MOIU.supports_default_copy_to(model::ConstrainedVariablesModel, ::Bool)
+function MOI.supports_incremental_interface(
+    model::ConstrainedVariablesModel,
+    ::Bool,
+)
     return !model.allocate_load
 end
 function MOIU.supports_allocate_load(model::ConstrainedVariablesModel, ::Bool)
@@ -226,7 +229,7 @@ function MOI.copy_to(
     return MOIU.automatic_copy_to(dest, src; kws...)
 end
 
-function MOIU.supports_default_copy_to(
+function MOI.supports_incremental_interface(
     model::AbstractConstrainedVariablesModel,
     ::Bool,
 )
@@ -581,7 +584,7 @@ function MOI.supports_constraint(
     return true
 end
 
-MOIU.supports_default_copy_to(::BoundModel, ::Bool) = true
+MOI.supports_incremental_interface(::BoundModel, ::Bool) = true
 function MOI.copy_to(dest::BoundModel, src::MOI.ModelLike; kws...)
     return MOIU.automatic_copy_to(dest, src; kws...)
 end