The relation on powersets associated to $R$ is the relation
\[ \mathcal{P}(R)\colon \mathcal{P}(X)\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}\mathcal{P}(Y) \]
defined as follows:1
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Viewing relations as functions to powersets, we have
\[ [\mathcal{P}(R)](U)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ V\in \mathcal{P}(Y)\ \middle |\ R(U)\cap V\neq \text{Ø}\right\} \]for each $U\in \mathcal{P}(X)$ and each $V\in \mathcal{P}(Y)$.
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Viewing relations as functions to $\{ \mathsf{t},\mathsf{f}\} $, we have
\[ \mathcal{P}(R)^{V}_{U}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathbf{Rel}(\Delta _{\mathrm{pt}},V\mathbin {\diamond }R\mathbin {\diamond }U) \]for each $U\in \mathcal{P}(X)$ and each $V\in \mathcal{P}(Y)$.
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1Illustration:
