The internal Hom of $\mathcal{P}\webleft (X\webright )$ from $U$ to $V$ is the subset $\webleft [U,V\webright ]_{X}$1 of $X$ defined by
\begin{align*} \webleft [U,V\webright ]_{X} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}\cup V\\ & = \webleft (U\setminus V\webright )^{\textsf{c}}\end{align*}
where $U^{\textsf{c}}$ is the complement of $U$ of Definition 2.3.11.1.1.
1Further Notation: Also written $\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )$.
