More specifically, if $\phi: \mathscr{F} \to \mathscr{G}$ is a morphism of sheaves and $\mathscr{G}' \subset \mathscr{G}$ is a subsheaf (so that $\mathscr{G}'(U) \subseteq \mathscr{G}(U)$ for all $U$ open), then what is the presheaf $U \mapsto \phi_U^{-1}(\mathscr{G}'(U))$? If they are $\mathscr{O}_X$-modules on a ringed space, do they enherit quasi-coherence from $\mathscr{G'}$? If $X$ is a noetherian scheme, can the same be said about coherence?
I was able to show it is a presheaf with the restriction maps induced by the ones on $\mathscr{F}$, but I have yet to show it is a sheaf or any of the coherence conditions. More than anything else, I am curious if this construction has a name so that I can learn more about it. It is hard to look up because the 'pullback sheaf' or 'inverse image sheaf' is a very common construction that is substantially different.
Thanks! I am going to continue working on this but I will include some progress below. I don't think it's necessary for my question so please feel free to ignore it if you don't need the context.
For a little more detail on the presheaf part, if $\rho: \mathscr{F}(U) \to \mathscr{F} (V)$ is a restriction then $\rho(\phi_U^{-1}(\mathscr{G}'(U))) \subseteq \phi_V^{-1}(\mathscr{G}'(V))$ by some diagram chasing so that $\rho|_{\phi_U^{-1}(\mathscr{G}'(U))}: \phi_U^{-1}(\mathscr{G}'(U)) \to \phi_V^{-1}(\mathscr{G}'(V))$ is a well defined map which satisfies the presheaf condition.
It is also a sheaf if $\mathscr{G}'$ is. As a sub-presheaf of $\mathscr{F}$, it inherits locality for free. For gluing, if we have $s_i \in \phi_{U_i}^{-1}(\mathscr{G}'(U_i))$ on an open cover $(U_i)$ of $U$ agreeing on intersections, they glue together to a section $s \in \mathscr{F}(U)$ since $\mathscr{F}$ is a sheaf. Then $\phi_{U_i}(s_i)$ also agree on intersections by the presheaf morphism condition so that they lift to a section of $\mathscr{G'}(U)$. By a locality argument on $\mathscr{G}$, it follows that this section is $\phi_U(s)$ so that $s \in \phi_U^{-1}(\mathscr{G}'(U))$ as required.
Checking the universal property, it is indeed isomorphic to $\mathscr{F} \times_{\mathscr{G}} \mathscr{G}'$ so this can be viewed as a fibered product.
This will be my last edit. If $X = \operatorname{Spec} A$ and $\tilde{M}, \tilde{N}$, and $\tilde{N'} \subseteq \tilde{N}$ are quasi-coherent sheaves, then a morphism $\phi: \tilde{M} \to \tilde{N}$ is induced by $\varphi: M \to N$, a morphism of $A$-modules. Then, it seems clear from the definitions that the preimage sheaf is just $\widetilde{\varphi^{-1}(N')}$, which is again quasi-coherent. I believe this settles the question of quasi-coherence of this sheaf on a scheme. By the same token, I suspect it is not necessarily coherent even if $\mathscr{G}'$ is, since preimages of finite modules are not always finite.