This question was inspired by reading about a criterion for a morphism into projective space (over an algebraically closed field) to be a closed immersion based on local rings. It got me thinking about how to frame this more generally. It should be an elementary question, but I am having trouble even formulating it properly.
Suppose we have a morphism of locally ringed spaces $(f, f^{\#}): (X, \mathcal{O}_{X}) \rightarrow (Y, \mathcal{O}_{Y})$. If $p \in X$ is a point, then we require that the morphism on stalks, $$ f^{\#}_{p}: \mathcal{O}_{Y, f(p)} \longrightarrow \mathcal{O}_{X, p} $$ be a local homomorphism of local rings. Heuristically, this means the following: If $s$ is a section of $\mathcal{O}_{Y}$ which vanishes in the residue field at $f(p)$, then the section "$s \circ f$" vanishes in the residue field at $p$.
How should this be generalised and interpreted for quasi-coherent sheaves, rather than just structure sheaves?
Let's restrict ourselves to proper morphisms (so quasi-coherence behaves nicely under morphisms) of noetherian schemes (so coherence plays nicely). I am happy to restrict even further if the question can be answered more clearly in a more strict setting.
Given a proper morphism $f: X \rightarrow Y$ of noetherian schemes. Let $\mathcal{F}$ be a coherent sheaf on $Y$. The first problem is what the corresponding morphism $f^{\#}$ should be for coherent sheaves, as opposed to the structure sheaf. I imagine it should be the unit of adjunction, $$ \eta: \mathcal{F} \longrightarrow f_{*}f^{*} \mathcal{F}. $$ How should one correctly interpret the idea that sections of $\mathcal{F}$ which vanish in the "residue vector space" at a point $f(p)$ in $Y$ pull back to sections which vanish in the "residue vector space" at $p$ in $X$?
My guess is that this is some direct application of Nakayama's lemma to the spaces $\mathcal{F}_{p} \otimes \kappa(p)$ but it is not clear to be how to even formulate the problem correctly. Can anyone offer some insight?