Projective objects in functor categories

Question

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).

Answer 1

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 $A\in R\text{-mod}$.

Proofs and more info can be found here.