• Since products in $\mathcal{P}\webleft (X\webright )$ are given by binary intersections (Item 1 of Proposition 2.4.1.1.4), the right adjoint $\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,-\webright )$ of $U\cap -$ may be thought of as a function type $\webleft [U,V\webright ]$.

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