Exactness of short sequences of chain complexes

Question

Given projective resolutions $P' \to A'$, $P \to A$ and $P'' \to A''$ in an abelian category $\mathcal{A}$ and a functor $F: \mathcal{A} \to \mathcal{B}$, where $\mathcal{B}$ is an abelian category, I came up to the following situation: For each $n$, the short sequence $$0 \to F(P'_{n}) \to F(P_{n}) \to F(P''_n) \to 0$$ is exact. From this point, can I assume that the short exact sequence of chain complexes $$0 \to F(P') \to F(P) \to F(P'') \to 0$$ is exact?