# 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?