Clarify definition of unique parent condition

src/sage/categories/homset.py

@@ -396,7 +396,7 @@ def Hom(X, Y, category=None, check=True):
     """
     # This should use cache_function instead
     # However some special handling is currently needed for
-    # domains/docomains that break the unique parent condition. Also,
+    # domains/codomains that break the unique parent condition. Also,
     # at some point, it somehow broke the coercion (see e.g. sage -t
     # sage.rings.real_mpfr). To be investigated.
     global _cache

src/sage/structure/coerce_dict.pyx

@@ -21,8 +21,7 @@ two parents (:issue:`715`) and to store homsets of pairs of objects of a
 category (:issue:`11521`). In both cases, it is essential that the parent
 structures remain garbage collectable, it is essential that the data access is
 faster than with a usual :class:`~weakref.WeakKeyDictionary`, and we enforce
-the "unique parent condition" in Sage (parent structures should be identical
-if they are equal).
+the "unique parent condition" in Sage.
 
 :class:`MonoDict` behaves similarly, but it takes a single item as a key. It
 is used for caching the parents which allow a coercion map into a fixed other

src/sage/structure/factory.pyx

@@ -658,7 +658,7 @@ def generic_factory_unpickle(factory, *args):
     (somewhere in the library of an old Sage version) a unique factory; in our
     example below, it returns polynomial rings. Now suppose that we want to
     replace the factory by something else, say, a class that provides the
-    unique parent behaviour using
+    unique representation behaviour using
     :class:`~sage.structure.unique_representation.UniqueRepresentation`. We
     show here how to make it possible to unpickle a pickle created with the
     factory, automatically turning it into an instance of the new class.

src/sage/structure/unique_representation.py

@@ -517,6 +517,9 @@ class that inherits from :class:`UniqueRepresentation`: By adding
 simultaneously inherit from :class:`CachedRepresentation` and from
 :class:`~sage.misc.fast_methods.WithEqualityById`.
 
+If the class is also a :class:`~sage.structure.parent.Parent`,
+then we says it satisfies the *unique parent* condition.
+
 For example, a symmetric function algebra is uniquely determined by the base
 ring. Thus, it is reasonable to use :class:`UniqueRepresentation` in this
 case::