The direct image with compact support function associated to $R$ is the function
\[ R_{!}\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]
\begin{align*} R_{!}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, if we have}\\ & \text{$b\in R\webleft (a\webright )$, then $a\in U$}\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ R^{-1}\webleft (b\webright )\subset U\webright\} \end{align*}
for each $U\in \mathcal{P}\webleft (A\webright )$.
1Further Terminology: The set $R_{!}\webleft (U\webright )$ is called the direct image with compact support of $U$ by $R$.
2We also have
\[ R_{!}\webleft (U\webright )=B\setminus R_{*}\webleft (A\setminus U\webright ); \]
see Item 7 of Proposition 7.4.4.1.3.
