Adjointness. We have adjunctions witnessed by bijections
\begin{align*} \mathsf{Sets}_{/X}\webleft (A\times _{X}B,C\webright ) & \cong \mathsf{Sets}_{/X}\webleft (A,\textbf{Sets}_{/X}\webleft (B,C\webright )\webright ),\\ \mathsf{Sets}_{/X}\webleft (A\times _{X}B,C\webright ) & \cong \mathsf{Sets}_{/X}\webleft (B,\textbf{Sets}_{/X}\webleft (A,C\webright )\webright ), \end{align*}
natural in $\webleft (A,\phi _{A}\webright ),\webleft (B,\phi _{B}\webright ),\webleft (C,\phi _{C}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{/X}\webright )$, where $\textbf{Sets}_{/X}\webleft (A,B\webright )$ is the object of $\mathsf{Sets}_{/X}$ consisting of (see 
, ):
- The Set. The set $\textbf{Sets}_{/X}\webleft (A,B\webright )$ defined by
\[ \textbf{Sets}_{/X}\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\coprod _{x\in X}\mathsf{Sets}\webleft (\phi ^{-1}_{A}\webleft (x\webright ),\phi ^{-1}_{Y}\webleft (x\webright )\webright ) \]
- The Map to $X$. The map
\[ \phi _{\textbf{Sets}_{/X}\webleft (A,B\webright )}\colon \textbf{Sets}_{/X}\webleft (A,B\webright )\to X \]
defined by
\[ \phi _{\textbf{Sets}_{/X}\webleft (A,B\webright )}\webleft (x,f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x \]
for each $\webleft (x,f\webright )\in \textbf{Sets}_{/X}\webleft (A,B\webright )$.