Descending Functions to Quotient Sets, III. Let $R$ be an equivalence relation on $X$. We have a bijection
\[ \textup{Hom}_{\mathsf{Sets}}\webleft (X/\mathord {\sim }_{R},Y\webright )\cong \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X,Y\webright ), \]
natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, given by the assignment $f\mapsto \overline{f}$ of Item 4 and Item 5, where $\textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X,Y\webright )$ is the set defined by
\[ \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X,Y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\in \textup{Hom}_{\mathsf{Sets}}\webleft (X,Y\webright )\ \middle |\ \begin{aligned} & \text{for each $x,y\in X$,}\\ & \text{if $x\sim _{R}y$, then}\\ & \text{$f\webleft (x\webright )=f\webleft (y\webright )$}\end{aligned} \webright\} . \]