Identifying subsets of $B$ with relations from $B$ to $\text{pt}$ via Chapter 2: Constructions With Sets, 
of , we see that the weak inverse image function associated to $R$ is equivalently the function
\[ R^{-1}\colon \underbrace{\mathcal{P}\webleft (B\webright )}_{\cong \mathrm{Rel}\webleft (B,\text{pt}\webright )}\to \underbrace{\mathcal{P}\webleft (A\webright )}_{\cong \mathrm{Rel}\webleft (A,\text{pt}\webright )} \]
defined by
\[ R^{-1}\webleft (V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}V\mathbin {\diamond }R \]
for each $V\in \mathcal{P}\webleft (A\webright )$, where $R\mathbin {\diamond }V$ is the composition
\[ A\mathbin {\overset {R}{\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}}}B \mathbin {\overset {V}{\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}}}\text{pt}. \]
Explicitly, we have
\begin{align*} R^{-1}\webleft (V\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}V\mathbin {\diamond }R\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\int ^{x\in B}V^{-_{1}}_{x}\times R^{x}_{-_{2}}. \end{align*}
