What are the restriction maps of the inverse image presheaf? Let $\pi:X \rightarrow Y$ be a continuous map of topological spaces and let $\mathscr{G}$ be a sheaf on $Y$. The inverse image presheaf $\pi^{-1}$ is defined as
$$ \pi^{-1}\mathscr{G}(U) = \lim\limits_{\substack{\longrightarrow \\ V \supseteq \pi(U)}} \mathscr{G}(V), $$
but usually the restriction maps are not specified. What are the restriction maps?
A definition I came up with is the following. Let $U \subseteq V$ be an inclusion of open sets. We have $\pi(U) \subseteq \pi(V)$, so every open set $W$ contanining $\pi(V)$, contains also $\pi(U)$. Therefore, we have maps $\mathscr{G}(W) \rightarrow \pi^{-1}\mathscr{G}(U)$. Finally, since $\pi^{-1}\mathscr{G}(V)$ is a colimit, the restriction map $\rho_{V,U}$ will be the unique map $\rho_{V,U}:\pi^{-1}\mathscr{G}(V) \rightarrow \pi^{-1}\mathscr{G}(U)$.
Is this the correct definition?
Why in many textbooks the restriction maps are not specified? (Is there some canonical way to deduce the restriction maps just from the definition of sections?)
 A: Yes, this is correct. You're using the fact that all of the objects in the diagram to calculate the value of $\pi^{-1}\mathscr{G}(V)$ are also objects in the diagram to calculate the value of $\pi^{-1}\mathscr{G}(U)$, and thus we have a map between the diagrams and thus a map between the limits.
As to why this isn't mentioned, it's because you don't need to check it often and if you have to, you can cook it up from the definition on sections without too much fuss just like you did. One big reason you might not think about this much is that $f^{-1}$ as a functor is not so common - usually, one deals with $f^*$, the composition of $f^{-1}$ and $-\otimes_{f^{-1}\mathcal{O}_Y}\mathcal{O}_X$, in order to get $\mathcal{O}_X$-modules out. With both $f^{-1}$ and $f^*$, needing to consider the specific form of a restriction map in order to make a proof work is not common. And even in the scenarios where you might need to consider such a thing, the fact that the map is induced by the properties of the inverse limit means that it's natural and thus easy to work with.

The big idea here about sheaves is that they're a tremendous amount of data, and we usually like to work with some sort of easier or less wordy representative (like when you call your friend - where I'm from, you usually just say their first name). For instance, when we talk about a quasicoherent sheaf on an affine scheme, we know that every such sheaf is of the form $\widetilde{M}$ for some module $M$. We hardly ever specify all of the restriction maps, even in this particularly easy case, because it would require us to say something about all the open sets. That's often difficult! Even in the Zariski topology, where there are far fewer open sets than the standard topology, we usually don't work explicitly with very many of our open sets.
