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The inverse image sheaf $f^{-1}\mathcal{E}$ is the sheaf associated with the presheaf whose sections near $U\subseteq X$ are obtained as a colimit of $\mathcal{E}(V)$ over open sets $V\supseteq f(U)$.

This is the sheaf obtained by applying the inverse image functor.

For a morphism of ringed spaces, the pullback of an ${\mathcal{O}}_Y$-module is

The inverse image functor supplies the sheaf $f^{-1}\mathcal{E}$; tensoring with the source structure sheaf supplies the module scalar extension.