Identifying subsets of $A$ with relations from $\text{pt}$ to $A$ via Chapter 2: Constructions With Sets, Item 3 of Proposition 2.4.3.1.6, we see that the direct image function associated to $R$ is equivalently the function
\[ R_{*}\colon \underbrace{\mathcal{P}\webleft (A\webright )}_{\cong \mathrm{Rel}\webleft (\text{pt},A\webright )} \to \underbrace{\mathcal{P}\webleft (B\webright )}_{\cong \mathrm{Rel}\webleft (\text{pt},B\webright )} \]
defined by
\[ R_{*}\webleft (U\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R\mathbin {\diamond }U \]
for each $U\in \mathcal{P}\webleft (A\webright )$, where $R\mathbin {\diamond }U$ is the composition
\[ \text{pt}\mathbin {\overset {U}{\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}}}A\mathbin {\overset {R}{\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}}}B. \]
