If $\{F_i: i\in I\}$ is a family of functors $F_i: \mathcal{A} \rightarrow \mathcal{B}$ between categories, then $\oplus_I F_i$ is exact if and only if each $F_i$ is exact?
What I tried:
Given a short exact sequence in R-mod $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ with non-trivial connecting morphisms $f,g$, and two functors $F, G$ applying $F \oplus G$ yields $$0 \rightarrow F(A) \oplus G(A) \rightarrow F(B) \oplus G(B) \rightarrow F(C)\oplus G(C) \rightarrow 0$$, with the maps $F(f) \oplus G(f)$ injective and $F(g)\oplus G(g)$ surjective, and $$Ker\ F(g) \oplus G(g) = Ker\ F(g) \times Ker\ G(g) = Im\ F(f) \times Im\ G(f) = Im\ F(f) \oplus G(f)$$ So we see that $F\oplus G$ is exact. Now this has limited scope obviously, but I guess I could generalize this to arbitrary categories and families of functors.
My question really is if the reverse is also true, ie. $\oplus_I F_i$ exact implies each $F_i$ exact?