Inverse-image direct-image adjunction
= Inverse-image direct-image adjunction
{title2=$f^{-1}\dashv f_*$}
For <sheaves of abelian groups> and a continuous map $f:X\to Y$, there is a natural bijection $\operatorname{Hom}_X(f^{-1}\mathcal G,\mathcal F)\cong\operatorname{Hom}_Y(\mathcal G,f_*\mathcal F)$. 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>.