Is it true that if I have exact sequences of abelian groups
$0 \rightarrow A\rightarrow B \rightarrow C\rightarrow 0$
and
$ A_1\rightarrow B_1 \rightarrow C_1 $ (exact only in the middle)
and morphisms
$a:A\rightarrow A_1$, $b:B\rightarrow B_1$ and $c:C\rightarrow C_1$ such that $a$ and $c$ are the zero morphism, then also $b$ is the zero morphism?
Thx