Item 3 (01LA) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

Naturality. The collection
\[ \webleft\{ \chi _{\webleft (-\webright )}\colon \mathcal{P}\webleft (X\webright )\to \mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright )\webright\} _{X\in \text{Obj}\webleft (\mathsf{Sets}\webright )} \]

defines a natural isomorphism between $\mathcal{P}^{-1}$ and $\mathsf{Sets}\webleft (-,\{ \mathsf{t},\mathsf{f}\} \webright )$. In particular, given a function $f\colon X\to Y$, the diagram

commutes, i.e. we have

\[ \chi _{V}\circ f=\chi _{f^{-1}\webleft (V\webright )} \]

for each $V\in \mathcal{P}\webleft (Y\webright )$.

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