f-morphism of sheaves ($\varphi (U):\mathcal{G}(U)\to \mathcal{F}(f^{-1}U)$)

For a continuous map $f:X\to Y$, an f-morphism from a sheaf on $Y$ to a sheaf on $X$ is a family of the displayed maps commuting with restrictions. It is equivalently a morphism of sheaves $\mathcal{G}\to f_{\ast }\mathcal{F}$. For each $x\in X$ it induces a map ${\mathcal{G}}_{f(x)}\to {\mathcal{F}}_x$ on stalks, by pulling back the neighbourhood representing a germ.