Proposition 2.1.3.1.3 (000U): Properties of Products of Sets

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Proposition 2.1.3.1.3 ▶ Properties of Products of Sets

Let $A$, $B$, $C$, and $X$ be sets.

Functoriality. The assignments $A,B,\webleft (A,B\webright )\mapsto A\times B$ define functors
\[ \begin{array}{ccc} A\times -\colon \mkern -15mu & \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ -\times B\colon \mkern -15mu & \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ -_{1}\times -_{2}\colon \mkern -15mu & \mathsf{Sets}\times \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}, \end{array} \]

where $-_{1}\times -_{2}$ is the functor where
- Action on Objects. For each $\webleft (A,B\webright )\in \text{Obj}\webleft (\mathsf{Sets}\times \mathsf{Sets}\webright )$, we have
  
  \[ \webleft [-_{1}\times -_{2}\webright ]\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A\times B. \]
- Action on Morphisms. For each $\webleft (A,B\webright ),\webleft (X,Y\webright )\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on $\textup{Hom}$-sets
  
  \[ \times _{\webleft (A,B\webright ),\webleft (X,Y\webright )} \colon \mathsf{Sets}\webleft (A,X\webright )\times \mathsf{Sets}\webleft (B,Y\webright )\to \mathsf{Sets}\webleft (A\times B,X\times Y\webright ) \]
  
  of $\times $ at $\webleft (\webleft (A,B\webright ),\webleft (X,Y\webright )\webright )$ is defined by sending $\webleft (f,g\webright )$ to the function
  
  \[ f\times g\colon A\times B\to X\times Y \]
  
  defined by
  
  \[ \webleft [f\times g\webright ]\webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f\webleft (a\webright ),g\webleft (b\webright )\webright ) \]
  
  for each $\webleft (a,b\webright )\in A\times B$.
and where $A\times -$ and $-\times B$ are the partial functors of $-_{1}\times -_{2}$ at $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Adjointness. We have adjunctions
witnessed by bijections
\begin{align*} \mathsf{Sets}\webleft (A\times B,C\webright ) & \cong \mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,C\webright )\webright ),\\ \mathsf{Sets}\webleft (A\times B,C\webright ) & \cong \mathsf{Sets}\webleft (B,\mathsf{Sets}\webleft (A,C\webright )\webright ), \end{align*}

natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Associativity. We have an isomorphism of sets
\[ \alpha ^{\mathsf{Sets}}_{A,B,C}\colon \webleft (A\times B\webright )\times C\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A\times \webleft (B\times C\webright ), \]

natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Unitality. We have isomorphisms of sets
\begin{align*} \lambda ^{\mathsf{Sets}}_{A} & \colon \text{pt}\times A \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A,\\ \rho ^{\mathsf{Sets}}_{A} & \colon A\times \text{pt}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A, \end{align*}

natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Commutativity. We have an isomorphism of sets
\[ \sigma ^{\mathsf{Sets}}_{A,B}\colon A\times B \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }B\times A, \]

natural in $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Distributivity Over Coproducts. We have isomorphisms of sets
\begin{align*} \delta ^{\mathsf{Sets}}_{\ell } & \colon A\times \webleft (B\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}C\webright ) \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (A\times B\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (A\times C\webright ),\\ \delta ^{\mathsf{Sets}}_{r} & \colon \webleft (A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B\webright )\times C \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\webleft (A\times C\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (B\times C\webright ), \end{align*}

natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Annihilation With the Empty Set. We have isomorphisms of sets
\begin{align*} \zeta ^{\mathsf{Sets}}_{\ell } & \colon \text{Ø}\times A \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\text{Ø},\\ \zeta ^{\mathsf{Sets}}_{r} & \colon A\times \text{Ø}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\text{Ø}, \end{align*}

natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Distributivity Over Unions. Let $X$ be a set. For each $U,V,W\in \mathcal{P}\webleft (X\webright )$, we have equalities
\begin{align*} U\times \webleft (V\cup W\webright ) & = \webleft (U\times V\webright )\cup \webleft (U\times W\webright ),\\ \webleft (U\cup V\webright )\times W & = \webleft (U\times W\webright )\cup \webleft (V\times W\webright ) \end{align*}

of subsets of $\mathcal{P}\webleft (X\times X\webright )$.
Distributivity Over Intersections. Let $X$ be a set. For each $U,V,W\in \mathcal{P}\webleft (X\webright )$, we have equalities
\begin{align*} U\times \webleft (V\cap W\webright ) & = \webleft (U\times V\webright )\cap \webleft (U\times W\webright ),\\ \webleft (U\cap V\webright )\times W & = \webleft (U\times W\webright )\cap \webleft (V\times W\webright ) \end{align*}

of subsets of $\mathcal{P}\webleft (X\times X\webright )$.
Distributivity Over Differences. Let $X$ be a set. For each $U,V,W\in \mathcal{P}\webleft (X\webright )$, we have equalities
\begin{align*} U\times \webleft (V\setminus W\webright ) & = \webleft (U\times V\webright )\setminus \webleft (U\times W\webright ),\\ \webleft (U\setminus V\webright )\times W & = \webleft (U\times W\webright )\setminus \webleft (V\times W\webright ) \end{align*}

of subsets of $\mathcal{P}\webleft (X\times X\webright )$.
Distributivity Over Symmetric Differences. Let $X$ be a set. For each $U,V,W\in \mathcal{P}\webleft (X\webright )$, we have equalities
\begin{align*} U\times \webleft (V\mathbin {\triangle }W\webright ) & = \webleft (U\times V\webright )\mathbin {\triangle }\webleft (U\times W\webright ),\\ \webleft (U\mathbin {\triangle }V\webright )\times W & = \webleft (U\times W\webright )\mathbin {\triangle }\webleft (V\times W\webright ) \end{align*}

of subsets of $\mathcal{P}\webleft (X\times X\webright )$.
Middle-Four Exchange with Respect to Intersections. The diagram

commutes, i.e. we have

\[ \webleft (U\times V\webright )\cap \webleft (W\times T\webright )=\webleft (U\cap V\webright )\times \webleft (W\cap T\webright ). \]

for each $U,V,W,T\in \mathcal{P}\webleft (X\webright )$.
Symmetric Monoidality. The 8-tuple $\webleft(\phantom{\mathrlap {\alpha ^{\mathsf{Sets}}}}\mathsf{Sets}\right.$, $\times $, $\text{pt}$, $\mathsf{Sets}\webleft (-_{1},-_{2}\webright )$, $\alpha ^{\mathsf{Sets}}$, $\lambda ^{\mathsf{Sets}}$, $\rho ^{\mathsf{Sets}}$, $\left.\sigma ^{\mathsf{Sets}}\webright)$ is a closed symmetric monoidal category.
Symmetric Bimonoidality. The 18-tuple
\[ \begin{aligned} & \webleft(\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }},\times ,\text{Ø},\text{pt},\mathsf{Sets}\webleft (-_{1},-_{2}\webright ),\alpha ^{\mathsf{Sets}},\lambda ^{\mathsf{Sets}},\rho ^{\mathsf{Sets}},\sigma ^{\mathsf{Sets}},\right.\\ & \left.\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}},\lambda ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}},\rho ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}},\sigma ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}},\delta ^{\mathsf{Sets}}_{\ell },\delta ^{\mathsf{Sets}}_{r},\zeta ^{\mathsf{Sets}}_{\ell },\zeta ^{\mathsf{Sets}}_{r}\webright),\end{aligned} \]

is a symmetric closed bimonoidal category, where $\alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$, $\lambda ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$, $\rho ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$, and $\sigma ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$ are the natural transformations from Item 2, Item 3, and Item 4 of Proposition 2.2.3.1.3.

Proof of Proposition 2.1.3.1.3.

Item 1: Functoriality

This follows from

Item 2: Adjointness

We prove only that there’s an adjunction $-\times B\dashv \mathsf{Sets}\webleft (B,-\webright )$, witnessed by a bijection

\[ \mathsf{Sets}\webleft (A\times B,C\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,C\webright )\webright ), \]

natural in $B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, as the proof of the existence of the adjunction $A\times -\dashv \mathsf{Sets}\webleft (A,-\webright )$ follows almost exactly in the same way.

Map I. We define a map

\[ \Phi _{B,C}\colon \mathsf{Sets}\webleft (A\times B,C\webright )\to \mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,C\webright )\webright ), \]

by sending a function

\[ \xi \colon A\times B\to C \]

to the function
where we define

\[ \xi ^{\dagger }_{a}\webleft (b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\xi \webleft (a,b\webright ) \]

for each $b\in B$. In terms of the $[\mspace {-3mu}[a\mapsto f\webleft (a\webright )]\mspace {-3mu}]$ notation of Chapter 1: Sets, Notation 1.1.1.1.2, we have

\[ \xi ^{\dagger }\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[a\mapsto [\mspace {-3mu}[b\mapsto \xi \webleft (a,b\webright )]\mspace {-3mu}]]\mspace {-3mu}]. \]
Map II. We define a map

\[ \Psi _{B,C}\colon \mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,C\webright )\webright ),\to \mathsf{Sets}\webleft (A\times B,C\webright ) \]

given by sending a function
to the function

\[ \xi ^{\dagger }\colon A\times B\to C \]

defined by

\begin{align*} \xi ^{\dagger }\webleft (a,b\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{ev}_{b}\webleft (\mathrm{ev}_{a}\webleft (\xi \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{ev}_{b}\webleft (\xi _{a}\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\xi _{a}\webleft (b\webright )\end{align*}

for each $\webleft (a,b\webright )\in A\times B$.
Invertibility I. We claim that

\[ \Psi _{A,B}\circ \Phi _{A,B}=\text{id}_{\mathsf{Sets}\webleft (A\times B,C\webright )}. \]

Indeed, given a function $\xi \colon A\times B\to C$, we have

\begin{align*} \webleft [\Psi _{A,B}\circ \Phi _{A,B}\webright ]\webleft (\xi \webright ) & = \Psi _{A,B}\webleft (\Phi _{A,B}\webleft (\xi \webright )\webright )\\ & = \Psi _{A,B}\webleft (\Phi _{A,B}\webleft ([\mspace {-3mu}[\webleft (a,b\webright )\mapsto \xi \webleft (a,b\webright )]\mspace {-3mu}]\webright )\webright )\\ & = \Psi _{A,B}\webleft ([\mspace {-3mu}[a\mapsto [\mspace {-3mu}[b\mapsto \xi \webleft (a,b\webright )]\mspace {-3mu}]]\mspace {-3mu}]\webright )\\ & = \Psi _{A,B}\webleft ([\mspace {-3mu}[a'\mapsto [\mspace {-3mu}[b'\mapsto \xi \webleft (a',b'\webright )]\mspace {-3mu}]]\mspace {-3mu}]\webright )\\ & = [\mspace {-3mu}[\webleft (a,b\webright )\mapsto \mathrm{ev}_{b}\webleft (\mathrm{ev}_{a}\webleft ([\mspace {-3mu}[a'\mapsto [\mspace {-3mu}[b'\mapsto \xi \webleft (a',b'\webright )]\mspace {-3mu}]]\mspace {-3mu}]\webright )\webright )]\mspace {-3mu}]\\ & = [\mspace {-3mu}[\webleft (a,b\webright )\mapsto \mathrm{ev}_{b}\webleft ([\mspace {-3mu}[b'\mapsto \xi \webleft (a,b'\webright )]\mspace {-3mu}]\webright )]\mspace {-3mu}]\\ & = [\mspace {-3mu}[\webleft (a,b\webright )\mapsto \xi \webleft (a,b\webright )]\mspace {-3mu}]\\ & = \xi . \end{align*}
Invertibility II. We claim that

\[ \Phi _{A,B}\circ \Psi _{A,B}=\text{id}_{\mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,C\webright )\webright )}. \]

Indeed, given a function
we have

\begin{align*} \webleft [\Phi _{A,B}\circ \Psi _{A,B}\webright ]\webleft (\xi \webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A,B}\webleft (\Psi _{A,B}\webleft (\xi \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A,B}\webleft ([\mspace {-3mu}[\webleft (a,b\webright )\mapsto \xi _{a}\webleft (b\webright )]\mspace {-3mu}]\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A,B}\webleft ([\mspace {-3mu}[\webleft (a',b'\webright )\mapsto \xi _{a'}\webleft (b'\webright )]\mspace {-3mu}]\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[a\mapsto [\mspace {-3mu}[b\mapsto \mathrm{ev}_{\webleft (a,b\webright )}\webleft ([\mspace {-3mu}[\webleft (a',b'\webright )\mapsto \xi _{a'}\webleft (b'\webright )]\mspace {-3mu}]\webright )]\mspace {-3mu}]]\mspace {-3mu}]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[a\mapsto [\mspace {-3mu}[b\mapsto \xi _{a}\webleft (b\webright )]\mspace {-3mu}]]\mspace {-3mu}]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[a\mapsto \xi _{a}]\mspace {-3mu}]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\xi .\end{align*}
Naturality for $\Phi $, Part I. We need to show that, given a function $g\colon B\to B'$, the diagram

commutes. Indeed, given a function

\[ \xi \colon A\times B'\to C, \]

we have

\begin{align*} \webleft [\Phi _{B,C}\circ \webleft (\text{id}_{A}\times g^{*}\webright )\webright ]\webleft (\xi \webright ) & = \Phi _{B,C}\webleft (\webleft [\text{id}_{A}\times g^{*}\webright ]\webleft (\xi \webright )\webright )\\ & = \Phi _{B,C}\webleft (\xi \webleft (-_{1},g\webleft (-_{2}\webright )\webright )\webright )\\ & = \webleft [\xi \webleft (-_{1},g\webleft (-_{2}\webright )\webright )\webright ]^{\dagger }\\ & = \xi ^{\dagger }_{-_{1}}\webleft (g\webleft (-_{2}\webright )\webright )\\ & = \webleft (g^{*}\webright )_{*}\webleft (\xi ^{\dagger }\webright )\\ & = \webleft (g^{*}\webright )_{*}\webleft (\Phi _{B',C}\webleft (\xi \webright )\webright )\\ & = \webleft [\webleft (g^{*}\webright )_{*}\circ \Phi _{B',C}\webright ]\webleft (\xi \webright ). \end{align*}

Alternatively, using the $[\mspace {-3mu}[a\mapsto f\webleft (a\webright )]\mspace {-3mu}]$ notation of Chapter 1: Sets, Notation 1.1.1.1.2, we have

\begin{align*} \webleft [\Phi _{B,C}\circ \webleft (\text{id}_{A}\times g^{*}\webright )\webright ]\webleft (\xi \webright ) & = \Phi _{B,C}\webleft (\webleft [\text{id}_{A}\times g^{*}\webright ]\webleft (\xi \webright )\webright )\\ & = \Phi _{B,C}\webleft (\webleft [\text{id}_{A}\times g^{*}\webright ]\webleft ([\mspace {-3mu}[\webleft (a,b'\webright )\mapsto \xi \webleft (a,b'\webright )]\mspace {-3mu}]\webright )\webright )\\ & = \Phi _{B,C}\webleft ([\mspace {-3mu}[\webleft (a,b\webright )\mapsto \xi \webleft (a,g\webleft (b\webright )\webright )]\mspace {-3mu}]\webright )\\ & = [\mspace {-3mu}[a\mapsto [\mspace {-3mu}[b\mapsto \xi \webleft (a,g\webleft (b\webright )\webright )]\mspace {-3mu}]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[a\mapsto g^{*}\webleft ([\mspace {-3mu}[b'\mapsto \xi \webleft (a,b'\webright )]\mspace {-3mu}]\webright )]\mspace {-3mu}]\\ & = \webleft (g^{*}\webright )_{*}\webleft ([\mspace {-3mu}[a\mapsto [\mspace {-3mu}[b'\mapsto \xi \webleft (a,b'\webright )]\mspace {-3mu}]]\mspace {-3mu}]\webright )\\ & = \webleft (g^{*}\webright )_{*}\webleft (\Phi _{B',C}\webleft ([\mspace {-3mu}[\webleft (a,b'\webright )\mapsto \xi \webleft (a,b'\webright )]\mspace {-3mu}]\webright )\webright )\\ & = \webleft (g^{*}\webright )_{*}\webleft (\Phi _{B',C}\webleft (\xi \webright )\webright )\\ & = \webleft [\webleft (g^{*}\webright )_{*}\circ \Phi _{B',C}\webright ]\webleft (\xi \webright ). \end{align*}
Naturality for $\Phi $, Part II. We need to show that, given a function $h\colon C\to C'$, the diagram

commutes. Indeed, given a function

\[ \xi \colon A\times B\to C, \]

we have

\begin{align*} \webleft [\Phi _{B,C}\circ h_{*}\webright ]\webleft (\xi \webright ) & = \Phi _{B,C}\webleft (h_{*}\webleft (\xi \webright )\webright )\\ & = \Phi _{B,C}\webleft (h_{*}\webleft ([\mspace {-3mu}[\webleft (a,b\webright )\mapsto \xi \webleft (a,b\webright )]\mspace {-3mu}]\webright )\webright )\\ & = \Phi _{B,C}\webleft ([\mspace {-3mu}[\webleft (a,b\webright )\mapsto h\webleft (\xi \webleft (a,b\webright )\webright )]\mspace {-3mu}]\webright )\\ & = [\mspace {-3mu}[a\mapsto [\mspace {-3mu}[b\mapsto h\webleft (\xi \webleft (a,b\webright )\webright )]\mspace {-3mu}]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[a\mapsto h_{*}\webleft ([\mspace {-3mu}[b\mapsto \xi \webleft (a,b\webright )]\mspace {-3mu}]]\mspace {-3mu}]\webright )\\ & = \webleft (h_{*}\webright )_{*}\webleft ([\mspace {-3mu}[a\mapsto [\mspace {-3mu}[b\mapsto \xi \webleft (a,b\webright )]\mspace {-3mu}]]\mspace {-3mu}]\webright )\\ & = \webleft (h_{*}\webright )_{*}\webleft (\Phi _{B,C}\webleft ([\mspace {-3mu}[\webleft (a,b\webright )\mapsto \xi \webleft (a,b\webright )]\mspace {-3mu}]\webright )\webright )\\ & = \webleft (h_{*}\webright )_{*}\webleft (\Phi _{B,C}\webleft (\xi \webright )\webright )\\ & = \webleft [\webleft (h_{*}\webright )_{*}\circ \Phi _{B,C}\webright ]\webleft (\xi \webright ). \end{align*}
Naturality for $\Psi $. Since $\Phi $ is natural in each argument and $\Phi $ is a componentwise inverse to $\Psi $ in each argument, it follows from Chapter 9: Categories, Item 2 of Proposition 9.9.7.1.2 that $\Psi $ is also natural in each argument.