Let $f:Y\to X$ be scheme morphism. $E\in QCoh(X)$. $L$ is line bundle over $Y$.
If $\phi: f^\star(E)\to L$ surjective, then $f^\star(E)\to f^\star f_\star(L)$ surjective.
$\textbf{Q:}$ Why is $f^\star(E)\to f^\star f_\star(L)$ obviously surjection which is induced by adjoint functoriality? I consider $Hom(f^\star E,L)=Hom(E,f_\star(L))$. Then $\phi\to\phi':E\to f_\star(L)$. Then $f^\star(\phi'):f^\star(E)\to f^\star f_\star(L)$. Then the question is to ask whether stalkwise is surjection. However this is asking $E_x\to f_\star(L)_x$ epimorphism. Now $f^\star(E)\to L$ gives $E_x\to L_y$ surjection for any $y\to x$. In particular, $E_x\to f_\star(L)_x$ is epimorphism. I found my reasoning very convoluted somehow and it is ad hoc. Is there an easy way to see $f^\star(E)\to f^\star f_\star(L)$?
$\textbf{Q':}$ I want to write down when $Y=P^1_k$ and $X=Spec(k)$, $f^\star(O_X(1))\to k^2$ identity, the total map $f^\star(V)\to f^\star(f_\star O_X(1))\to O_X(1)$ concretely. Is there an easy way to write it down?