Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be a functor of abelian categories. We will define some properties of $F$ before we state a question.
Let $X \rightarrow Y \rightarrow Z$ be a sequence in $\mathcal{A}$. We say $F$ reflects exact sequences if $X \rightarrow Y \rightarrow Z$ is exact whenever $F(X) \rightarrow F(Y) \rightarrow F(Z)$ is exact.
We say $F$ is faithfully exact if it is exact and reflects exact sequences.
Let $X$ be an object of $\mathcal{A}$. We say $F$ reflects zero objects if $X = 0$ whenever $F(X) = 0$.
Let $f\colon X \rightarrow Y$ be a morphism of $\mathcal{A}$. We say $F$ reflects zero morphisms if $f = 0$ whenever $F(f) = 0$.
Question. Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be an exact functor of abelian categories. Are the following conditions equivalent?
(1) $F$ reflects exact sequences, i.e., $F$ is faithfully exact.
(2) Let $f\colon X \rightarrow Y$ and $g\colon Y \rightarrow Z$ be morphisms of $\mathcal{A}$ such that $gf = 0$. Suppose $F(X) \rightarrow F(Y) \rightarrow F(Z)$ is exact. Then $X \rightarrow Y \rightarrow Z$ is exact.
(3) $F$ reflects zero objects.
(4) $F$ reflects zero morphisms.