Inverse-image direct-image adjunction ($f^{-1}⊣f_{\ast }$)

For sheaves of abelian groups and a continuous map $f:X\to Y$, there is a natural bijection ${\mathrm{Hom}}_X(f^{-1}\mathcal{G},\mathcal{F})\cong {\mathrm{Hom}}_Y(\mathcal{G},f_{\ast }\mathcal{F})$. It comes from the universal property of an inverse image sheaf. The inverse image of abelian sheaves is different from the tensor-adjusted pullback of a sheaf of modules on a ringed space.