Let $A$ and $B$ be two algebras over a field $\mathbb K$. $F\colon \mod A \rightarrow \mod B$ is a convariant functor. Let $S$ be a module in $\mod A$ such that $F(S)=0$. Then the trivial homomorphism $u\colon 0 \rightarrow S$ in $\mod A$ induces an isomorphism $F(u):F(0) \rightarrow F(S)$.
Then can we get $u$ an isomorphism by $F$ faithful?