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.