Let $f : (X ,\mathcal{O}_X) \to (Y ,\mathcal{O}_Y)$ be a map of ringed spaces. Let $\mathcal{G}$ be a sheaf on $Y$. How does one go about proving that $f^{-1}\mathcal{G}$ is an $f^{-1}\mathcal{O}_Y$-module?
There is this annoying sheafification process which one needs to go over in order to prove this and I'm at lost trying to sort out the technicalities.
The presheaves we consider are say $f^{-1}\mathcal{G}^{\text{pre}}$ and $f^{-1}\mathcal{O}_Y^{\text{pre}}$ with
$$ \begin{align*} f^{-1}\mathcal{G}^{\text{pre}}(U) &= \lim_{V \supset f(U)} \mathcal{G}(V) \\ f^{-1}\mathcal{O}_Y^{\text{pre}}(U) &= \lim_{V \supset f(U)} \mathcal{O}_Y(V). \end{align*} $$
Is it enough to do some magic here with the presheaves and show that $f^{-1}\mathcal{G}^{\text{pre}}$ is an $f^{-1}\mathcal{O}_Y^{\text{pre}}$-module? Maybe using the universality of sheafification and some properties of the limit?
It is important to stress that $$ \begin{align*} f^{-1}\mathcal{G}^{\text{pre}}(U) &= \mathrm{colim}_{V \supset f(U)}\, \mathcal{G}(V) \\ f^{-1}\mathcal{O}_Y^{\text{pre}}(U) &= \mathrm{colim}_{V \supset f(U)}\, \mathcal{O}_Y(V). \end{align*} $$ so we don't have a limit but a colimit (sometimes also called a direct limit, confusingly).
Now, these are filtered colimits, and filtered colimits in sets preserve finite limits. As (co)limits in presheaf categories are computed objectwise, filtered colimits of presheaves also commute with finite limits of presheaves. Therefore, the ''presheaf inverse image'' functor preserves finite limits, in particular products: $f^{-1}(\mathcal{G}\times\mathcal{H})^\mathrm{pre}\cong f^{-1}\mathcal{G}^\mathrm{pre}\times f^{-1}\mathcal{G}^\mathrm{pre}$. Sheafification also preserves finite limits, which is a standard fact that is worth remembering. Therefore the functor $f^{-1}\colon\mathrm{Sh}(Y)\to\mathrm{Sh}(X)$ preserves finite limits, in particular products.
It is now just formal reasoning that $f^{-1}$ preserves modules: ring objects are categorically described in terms of certain commutative diagrams involving categorical products, and module objects over ring objects are defined by further commutative diagrams involving categorical products. Since $f^{-1}$ preserves products (and commutativity of diagrams because it is a functor), the functor $f^{-1}$ preserves ring and module objects. As such, if $\mathcal{M}$ is an $\mathcal{O}_Y$-module, then $f^{-1}\mathcal{O}_Y$ is a ring object in $\mathrm{Sh}(X)$ and $f^{-1}\mathcal{M}$ is an $f^{-1}\mathcal{O}_Y$-module.