Projective

Let be a class of Epimorphisms in a Locally Small Category
An Object in is projective
if Hom-Functor preserves Epimorphisms in
i.e. is surjective for all in .
i.e. if given

where is an Epimorphism in
then there exists with

We say that is projective if are all Epimorphisms in

Lemma

For any Small Category , the functors are pointwise projective in

Proof

Let and some Functors.
Suppose is a Natural Transformation
and that is surjective.
Let be a Natural Transformation.
By Yoneda Lemma, corresponds to some (i.e. )
Then there is some such that
But then there is some Natural Transformation
such that (again by Yoneda Lemma)
We conclude that

But is also a Natural Transformation so we can write

and is bijective so

Proposition

Let be a Small Category.
For any Functor
there is a pointwise Epimorphism in
with being pointwise projective.

Proof

Given , let be the disjoint union

where the union is taken over and .
Now is a Coproduct
A Coproduct of -projective objects is -projective so is pointwise projective.
The morphism whose th component

is pointwise Epimorphism, since is in the image of .