The characteristic embedding1 of $X$ into $\mathcal{P}\webleft (X\webright )$ is the function
\[ \chi _{\webleft (-\webright )}\colon X \hookrightarrow \mathcal{P}\webleft (X\webright ) \]
defined by2
\begin{align*} \chi _{\webleft (-\webright )}\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{x}\\ & = \webleft\{ x\webright\} \end{align*}
for each $x\in X$.
1The name “characteristic embedding” is justified by Corollary 2.5.5.1.2, which gives an analogue of fully faithfulness for $\chi _{\webleft (-\webright )}$.
2Here we are identifying $\mathcal{P}\webleft (X\webright )$ with $\mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright )$ as per Item 2 of Proposition 2.5.1.1.4.
