diff --git a/src/sage/schemes/generic/algebraic_scheme.py b/src/sage/schemes/generic/algebraic_scheme.py
index df42dc0c324..e9bdfd0cdd1 100644
--- a/src/sage/schemes/generic/algebraic_scheme.py
+++ b/src/sage/schemes/generic/algebraic_scheme.py
@@ -130,7 +130,6 @@
 #          class AlgebraicScheme_subscheme_affine_toric
 #    class AlgebraicScheme_quasi
 
-from copy import copy
 from sage.categories.number_fields import NumberFields
 
 from sage.rings.all import ZZ
@@ -2295,7 +2294,7 @@ def orbit(self, f, N):
             Orb.append(Q)
         return(Orb)
 
-    def nth_iterate(self, f, n , **kwds):
+    def nth_iterate(self, f, n):
         r"""
         The nth forward image of this scheme by the map ``f``.
 
@@ -2354,15 +2353,10 @@ def nth_iterate(self, f, n , **kwds):
             ...
             TypeError: Attempt to coerce non-integral RealNumber to Integer
         """
-        if not f.is_endomorphism():
-            raise TypeError("map must be an endomorphism for iteration")
         n = ZZ(n)
         if n < 0:
             raise TypeError("must be a forward orbit")
-        Q = self
-        for i in range(n):
-            Q = f(Q)
-        return Q
+        return self.orbit(f,[n,n+1])[0]
 
     def _forward_image(self, f):
         """
@@ -2574,9 +2568,7 @@ def preimage(self, f):
         if self.ambient_space() != codom:
             raise TypeError("subscheme must be in ambient space of codomain")
         R = codom.coordinate_ring()
-        dict = {}
-        for i in range(codom.dimension_relative()+1):
-            dict.update({R.gen(i): f[i]})
+        dict = {R.gen(i): f[i] for i in range(codom.dimension_relative()+1)}
         return(dom.subscheme([t.subs(dict) for t in self.defining_polynomials()]))
 
 class AlgebraicScheme_subscheme_product_projective(AlgebraicScheme_subscheme_projective):
diff --git a/src/sage/schemes/projective/projective_point.py b/src/sage/schemes/projective/projective_point.py
index 38f2951bc5d..c4a8fa2dedd 100644
--- a/src/sage/schemes/projective/projective_point.py
+++ b/src/sage/schemes/projective/projective_point.py
@@ -683,14 +683,10 @@ def nth_iterate(self,f, n, **kwds):
             sage: P2.<u,v,w> = ProjectiveSpace(ZZ,2)
             sage: H = Hom(P,P2)
             sage: f = H([x^2+3*y^2,2*y^2,z^2])
-            sage: P2(2,7,1).nth_iterate(f,2)
-            Traceback (most recent call last):
-            ...
-            TypeError: point is not defined over domain of function
             sage: P(2,7,1).nth_iterate(f,2)
             Traceback (most recent call last):
             ...
-            TypeError: map must be an endomorphism
+            TypeError: map must be an endomorphism for iteration
 
         ::
 
@@ -716,23 +712,10 @@ def nth_iterate(self,f, n, **kwds):
 
         .. TODO:: Is there a more efficient way to do this?
         """
-        if self.codomain() != f.domain():
-            raise TypeError("point is not defined over domain of function")
-        if not f.is_endomorphism():
-            raise TypeError("map must be an endomorphism")
         n = ZZ(n)
         if n < 0:
             raise TypeError("must be a forward orbit")
-        Q = self
-        normalize = kwds.pop("normalize", False)
-        check = kwds.pop("check",True)
-        if normalize:
-            Q.normalize_coordinates()
-        for i in range(n):
-            Q = f(Q, check)
-            if normalize:
-                Q.normalize_coordinates()
-        return(Q)
+        return self.orbit(f,[n,n+1],**kwds)[0]
 
     def orbit(self, f, N, **kwds):
         r"""
@@ -849,7 +832,7 @@ def orbit(self, f, N, **kwds):
         if N[0] > N[1]:
             return([])
 
-        Q = copy(self)
+        Q = self
         check = kwds.pop("check",True)
         normalize = kwds.pop("normalize",False)