The Clowder Project

\begin{align*} f_{!,\text{im}}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (U\webright )\cap \mathrm{Im}\webleft (f\webright )\\ & = \webleft\{ y\in Y\ \middle |\ \text{we have $f^{-1}\webleft (y\webright )\subset U$ and $f^{-1}\webleft (y\webright )\neq \text{Ø}$}\webright\} .\end{align*}