The bijection
\[ \mathcal{P}\webleft (X\webright )\cong \mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright ) \]
of Item 1 of Proposition 2.4.3.1.6, which
- Takes a subset $U\hookrightarrow X$ of $X$ and straightens it to a function $\chi _{U}\colon X\to \{ \mathsf{true},\mathsf{false}\} $;
- Takes a function $f\colon X\to \{ \mathsf{true},\mathsf{false}\} $ and unstraightens it to a subset $f^{-1}\webleft (\mathsf{true}\webright )\hookrightarrow X$ of $X$;
\[ \underbrace{\mathsf{FibSets}\webleft (X\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{Sets}_{/X}}\cong \underbrace{\mathsf{ISets}\webleft (X\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathsf{Fun}\webleft (X_{\mathsf{disc}},\mathsf{Sets}\webright )} \]
of 
, where we view:
- Subsets $U\hookrightarrow X$ as analogous to $X$-fibred sets $\phi _{X}\colon A\to X$.
- Functions $f\colon X\to \{ \mathsf{t},\mathsf{f}\} $ as analogous to $X$-indexed sets $A\colon X_{\mathsf{disc}}\to \mathsf{Sets}$.
