Page "Equivalence of categories" Paragraph 3

An equivalence of categories consists of a functor between the involved categories, which is required to have an " inverse " functor.

However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its " inverse " is not necessarily the identity mapping.

Instead it is sufficient that each object be naturally isomorphic to its image under this composition.

Thus one may describe the functors as being " inverse up to isomorphism ".

There is indeed a concept of isomorphism of categories where a strict form of inverse functor is required, but this is of much less practical use than the equivalence concept.