Let $X, Y$ be topological spaces, $\mathcal{G}$ a sheaf on $Y$. We define $f^{-1}\mathcal{G}$ as the sheaf associated to the presheaf $f^+\mathcal{G}(U) = \underset{V \supseteq f(U)}{\varinjlim}\mathcal{G}(V)$. The idea is to approximate $f(U)$ by open sets containing it - gluing sections that "agree on $f(U)$". I was thinking in maybe another way to do it. Instead of approximating by sets containing $U$, we could approximate by sets contained in it.
What motivated me is the following fact: given a basis $\mathfrak{B}$ on a topological space $X$, and a sheaf on a basis $F$, we can define a sheaf $\mathcal{F}(U) = \underset{B \in \mathfrak{B}, B \subseteq U}{\varprojlim}F(B)$. I thought that we could define $f^{-1}\mathcal{G}$ as the sheaf associated to $U \mapsto \underset{V \subseteq f(U)}{\varprojlim}\mathcal{G}(V)$. Would that work?