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[Merged by Bors] - feat(category_theory/opposites): iso.unop #7400

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13 changes: 13 additions & 0 deletions src/category_theory/opposites.lean
Original file line number Diff line number Diff line change
Expand Up @@ -271,6 +271,19 @@ protected def op (α : X ≅ Y) : op Y ≅ op X :=
hom_inv_id' := quiver.hom.unop_inj α.inv_hom_id,
inv_hom_id' := quiver.hom.unop_inj α.hom_inv_id }

/-- The isomorphism obtained from an isomorphism in the opposite category. -/
@[simps] def unop {X Y : Cᵒᵖ} (f : X ≅ Y) : (Y.unop ≅ X.unop) :=
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{ hom := f.hom.unop,
inv := f.inv.unop,
hom_inv_id' := by simp only [← unop_comp, f.inv_hom_id, unop_id],
inv_hom_id' := by simp only [← unop_comp, f.hom_inv_id, unop_id] }

@[simp] lemma unop_op {X Y : Cᵒᵖ} (f : X ≅ Y) : f.unop.op = f :=
by ext; refl

@[simp] lemma op_unop {X Y : C} (f : X ≅ Y) : f.op.unop = f :=
by ext; refl

end iso

namespace nat_iso
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