Let $p_A:A\longrightarrow M$ and $p_B:B\longrightarrow N$ be vector bundles and $\Phi:A\longrightarrow B$ a vector bundle morphism covering $\phi:M\longrightarrow N$.
For $p\in M$, define $\Phi_p:=\Phi|_{A_p}:A_p\longrightarrow B_{\phi(p)}$ where $A_p:=p_A^{-1}(p)$ and $B_{\phi(p)}:=p_B^{-1}(\phi(p))$.
Is it true that:
$(i)$ $\Phi$ is surjective $\Leftrightarrow$ $\Phi_p$ is surjective for all $p\in M$;
$(ii)$ $\Phi$ is injective $\Leftrightarrow$ $\Phi_p$ is injective for all $p\in M$;
$(iii)$ $\Phi$ is bijective $\Leftrightarrow$ $\Phi_p$ is bijective for all $p\in M$?
If that is not true then:
$(a)$ which implications do hold?
$(b)$ what would be hypothesis on $\Phi$ or $\phi$ to make it true?
(i) and (iii) are not true since $\phi$ can be not surjective. Example. Take the trivials bundle $R\times M\rightarrow M$ and $R\times (M+M)\rightarrow M+M$. Where $M+M$ is the disjoint copy of two examples of $M$.