Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}$ define functors
\[ \begin{array}{ccc} {\webleft [U,-\webright ]_{X}}\colon \mkern -15mu & {\webleft (\mathcal{P}\webleft (X\webright ),\supset \webright )} \mkern -17.5mu& {}\mathbin {\to }{\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )},\\ {\webleft [-,V\webright ]_{X}}\colon \mkern -15mu & {\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )} \mkern -17.5mu& {}\mathbin {\to }{\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )},\\ {\webleft [-_{1},-_{2}\webright ]_{X}}\colon \mkern -15mu & {\webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \supset \webright )} \mkern -17.5mu& {}\mathbin {\to }{\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )}. \end{array} \]
In particular, the following statements hold for each $U,V,A,B\in \mathcal{P}\webleft (X\webright )$:
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If $U\subset A$, then $\webleft [A,V\webright ]_{X}\subset \webleft [U,V\webright ]_{X}$.
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If $V\subset B$, then $\webleft [U,V\webright ]_{X}\subset \webleft [U,B\webright ]_{X}$.
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If $U\subset A$ and $V\subset B$, then $\webleft [A,V\webright ]_{X}\subset \webleft [U,B\webright ]_{X}$.