algebraic geometry – Is the tensor by an invertible sheaf functor fully faithful?


My question arises while trying to prove that given a coherent $mathcal{O}_X$-module $F$ and an invertible locally free sheaf of rank $1$ (line bundle) $L$, then $H^0(X,Fotimes L^n)cong Hom(L^{-n},F).$

Here $L^n$ denotes the $n$-th tensor power of $L$, and the minus sign indicates its inverse in the Picard group of $X$. I’d like to prove that using the tensor-Hom adjunction and other two facts that maybe very simple but I’m not so sure about. My attempt would be so to consider isomorphisms
$$Hom(-,Hom(L^{-n},F))cong Hom(L^{-n}otimes -,F)cong Hom(-,L^notimes F)cong Hom(-,H^0(X,L^notimes F)),$$
and eventually use Yoneda embedding. Are the two last step correct? That is $(i)$ tensor product functor $Lotimes -$ is fully faithful and $(ii)$ there is isomorphism between groups $Hom_{mathcal{O_{X}}}(-,G)$ and $Hom_{Ab}(-,G(X))$ for a sheaf of modules $G$? If so, how to prove them? If not, how to prove the original statement?

Really thanks in advance 🙂