Global Section as Sheaf Morphism

Question

Let $X$ be scheme, $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Obviously $\mathcal{L}^{\otimes n}$ is invertible too.

Let consider an arbitrary global section $s \in \mathcal{L}^{\otimes n}(X)$. In Hartshorne $s$ was one time called a "sheaf morphism" $s:\mathcal{O}_X \to \mathcal{L}^{\otimes n}$. In which sense a global section can be understood as sheaf morphism?

Answer 1

We have $\operatorname{Hom}_{\mathcal O_X}(\mathcal O_X,-) \cong \Gamma(X,-)$, i.e. giving a sheaf morphism from $\mathcal O_X$ is the same as giving a global section.

You can either directly establish a natural isomorphism $\operatorname{Hom}(\mathcal O_X,\mathcal F) \cong \Gamma(X,\mathcal F)$ to show the isomorphism (You basically globalize the well known $\operatorname{Hom}_R(R,M)=M$) of functors or you can do some abstract nonsense to globalize it:

A scheme comes with a natural morphism $f:X \to \operatorname{Spec} \Gamma(X,\mathcal O_X)=:Y$.

We have $$\operatorname{Hom}_{\mathcal O_X}(\mathcal O_X,-)=\operatorname{Hom}_{\mathcal O_X}(f^*\mathcal O_Y,-)=\operatorname{Hom}_{\mathcal O_Y}(\mathcal O_Y,f_*(-))\\=\operatorname{Hom}_{\mathcal O_Y}(\mathcal O_Y,\widetilde{\Gamma(X,-)})=\operatorname{Hom}_{\Gamma(X,\mathcal O_X)}(\Gamma(X,\mathcal O_X),\Gamma(X,-)) = \Gamma(X,-).$$