I am trying to understand the idea of colimit used on the definition of inverse image phesheaf: let $f:X\to Y$ be a continuous function of topological spaces and $G$ a sheaf on $Y$, define a presheaf on $X$ associating every open $U\subset X$ to
$(f^{-1}G)(U)=colimit\{G(V),f(U)\subset V\}$
Can I think about the $colimit\{G(V),f(U)\subset V\}$ as $G(V)$ where $V$ is the "limit" (the smaller) open $V$ on $Y$ that contains $f(U)$?
I am having problems to catch the idea of colimit in general and specifically in this case.
Sorry for the colimit word, I look for a command but did not found one.
I appreciate any help.
The way I would recommend to think about this colimit is that an element of $(f^{-1}G)(U)$ is an element of $G(V)$ where $V$ is an "arbitrarily small" open neighborhood of $f(U)$. More precisely, an element of $(f^{-1}G)(U)$ is given by an element of $G(V)$ where $V$ is an open neighborhood of $f(U)$. These elements are then subject to the equivalence relation that $x\in G(V)$ and $y\in G(V')$ represent the same element of $(f^{-1}G)(U)$ if there exists an open neighborhood $V''\subseteq V\cap V'$ of $U$ such that the restrictions of $x$ and $y$ to $V''$ are equal. This equivalence relation is where the idea of "arbitrarily small" comes from: if two sections become equal in a small enough neighborhood of $f(U)$, we consider them to be equal.