The Clowder Project

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