2.5.5 The Yoneda Lemma for Sets
Let $X$ be a set and let $U\subset X$ be a subset of $X$.
We have
\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{U}\webright )=\chi _{U}\webleft (x\webright ) \]
for each $x\in X$, giving an equality of functions
\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{\webleft (-\webright )},\chi _{U}\webright )=\chi _{U}, \]
where
\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $U\subset V$,}\\ \mathsf{false}& \text{otherwise.} \end{cases} \]
We have
\begin{align*} \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{U}\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $\webleft\{ x\webright\} \subset U$,}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & = \begin{cases} \mathsf{true}& \text{if $x\in U$}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{U}\webleft (x\webright ). \end{align*}
This finishes the proof.
The characteristic embedding is fully faithful, i.e., we have
\[ \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{y}\webright )\cong \chi _{X}\webleft (x,y\webright ) \]
for each $x,y\in X$.
We have
\begin{align*} \chi _{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{y}\webright ) & = \chi _{y}\webleft (x\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $x\in \webleft\{ y\webright\} $}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & = \begin{cases} \mathsf{true}& \text{if $x=y$}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{X}\webleft (x,y\webright ). \end{align*}
where we have used Proposition 2.5.5.1.1 for the first equality.