Proposition 2.4.3.1.1 (01JG): Adjointness of Powersets I

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We have an adjunction

witnessed by a bijection

\[ \underbrace{\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (X\webright ),Y\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mkern 5mu\mathsf{Sets}\webleft (Y,\mathcal{P}\webleft (X\webright )\webright )} \cong \mathsf{Sets}\webleft (X,\mathcal{P}\webleft (Y\webright )\webright ), \]

natural in $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $Y\in \text{Obj}\webleft (\mathsf{Sets}^{\mathsf{op}}\webright )$.

We have

$\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (A\webright ),B\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\rlap {\mathsf{Sets}\webleft (B,\mathcal{P}\webleft (A\webright )\webright )}\phantom{\mkern 400mu}$

$\phantom{\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (X\webright ),Y\webright )}\cong \rlap {\mathsf{Sets}\webleft (B,\mathsf{Sets}\webleft (A,\{ \mathsf{t},\mathsf{f}\} \webright )\webright )}\phantom{\mkern 400mu}$

(by Item 2 of Proposition 2.5.1.1.4)

$\phantom{\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (X\webright ),Y\webright )} \cong \rlap {\mathsf{Sets}\webleft (A\times B,\{ \mathsf{t},\mathsf{f}\} \webright )}\phantom{\mkern 400mu}$

(by Item 2 of Proposition 2.1.3.1.3)

$\phantom{\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (X\webright ),Y\webright )} \cong \rlap {\mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,\{ \mathsf{t},\mathsf{f}\} \webright )\webright )}\phantom{\mkern 400mu}$

(by Item 2 of Proposition 2.1.3.1.3)

$\phantom{\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (X\webright ),Y\webright )} \cong \rlap {\mathsf{Sets}\webleft (A,\mathcal{P}\webleft (B\webright )\webright )}\phantom{\mkern 400mu}$

(by Item 2 of Proposition 2.5.1.1.4)

with all bijections natural in $A$ and $B$.¹

¹Here we are using Item 3 of Proposition 2.5.1.1.4.