Projective objects in functor categories

Let $$A$$ be an abelian category and $$I$$ some arbitrary category. It follows, that the functor category $$A^I$$ is also an abelian category.

Is there a general characterization of the projective objects of $$A^I$$ in terms of the projective objects of $$A$$? If not, how much structure for $$I$$ do we have to require to obtain nice results?

For example, the complex category $$\text{Comp}(A)$$ can be understood as a category of additive functors according to this answer and there is a characterization of projective chain complexes in terms of their underlying objects, cycles, boundaries and homology (they must all be projective).

• I think that's a very hard question. For instance, if you take $A=\mathbf{Ab}$, you have special projectives that are $\mathbb Z[\hom(i,-)], i\in I$, and these are generators so any projective functor is a summand of one of these - but said summands can be wild. Another point of view on why it's a hard question is to take $I=BG$, for $G$ a group. Then you're looking at the category of $G$-representations in $A$, and projectives in that one can be weird too (although in some cases you can determine them) Commented Apr 1, 2020 at 17:36

Here is a partial answer when the target category is the category $$\textbf{Ab}$$ of abelian groups.
Lemma. If $$\mathcal{C}$$ is a small preadditive category then the finitely generated projective objects in $$(\mathcal{C},\textbf{Ab})$$ are the finite direct summands of direct sums of representable functors. If $$\mathcal{C}$$ has split idempotents and finite direct sums then these are precisely the representable functors.
Recall that a functor is representable if it is of the form $$\text{Hom}(C,-)$$ for some $$C\in\mathcal{C}$$.
For example, if $$R$$ is a ring then the finitely generated projective objects in $$(R\text{-mod},\textbf{Ab})$$ are of the form $$\text{Hom}_{R}(A,-)$$ for $$A\in R\text{-mod}$$.