Essentially Injective

Let be a Functor Then is essentially injective if it is injective on Isomorphism classes i.e. suppose is an Isomorphism Then there is an Isomorphism and: Suppose that given an isomorphism the unique with is an isomorphism, with inverse the unique with

Lemma

If is Full and Faithfull, then is essentially injective

Proof

Let be Full and Faithfull. Let be an Isomorphism in As is Full, there is some Morphism with as well as a Morphism with Moreover, as is Faithfull and

then

and thus is an Isomorphism. Even stronger, as is Faithfull, is unique.