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For a continuous map $f:X\to Y$ and a sheaf $\mathcal{F}$ on $X$, the direct image is defined by $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$ for every open $V\subseteq Y$.

The direct image sheaf is the result of the direct image functor: $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}V)$.

For a continuous map $f:X\to Y$, the inverse-image functor takes a sheaf on $Y$ to the sheafification of its local inverse-image presheaf on $X$. It is left adjoint to the direct image functor. For sheaves of modules on ringed spaces, extension of scalars gives the separate pullback of a sheaf of modules.