First isomorphism theorem of sheaves -- do you need to sheafify if the map is surjective on basis sets?

Question

Say I have two sheaves $\mathscr F, \mathscr G$ of rings over a topological space $X$, and basis set $\mathscr B \subset \mathscr P(X)$. Say I've specified a morphism $\mathscr F \rightarrow \mathscr G$ over elements of $\mathscr B$, such that each $\mathscr F(U) \rightarrow \mathscr G(U)$ is surjective ($U \in \mathscr B$). I think this is enough to specify a surjective morphism $\phi: \mathscr F \rightarrow \mathscr G$, right? Then I can take the quotient presheaf $\mathscr F / ker(\phi)$. But this presheaf is now isomorphic to $\mathscr G$, and is therefore a sheaf. Is that correct?