of $\mathsf{Sets}$ admits an internal Hom $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}}$.
The Unit Object Is $\text{pt}$. We have $\mathbb {1}_{\mathsf{Sets}}=\text{pt}$.
More precisely, the full subcategory of the category $\mathcal{M}^{\mathrm{cld}}_{\mathbb {E}_{\infty }}\webleft (\mathsf{Sets}\webright )$ of spanned by the closed symmetric monoidal categories $\webleft(\phantom{\mathrlap {\lambda ^{\mathsf{Sets}}}}\mathsf{Sets}\right.$, $\otimes _{\mathsf{Sets}}$, $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}}$, $\mathbb {1}_{\mathsf{Sets}}$, $\lambda ^{\mathsf{Sets}}$, $\rho ^{\mathsf{Sets}}$, $\left.\sigma ^{\mathsf{Sets}}\webright)$ satisfying Item 1 and Item 2 is contractible (i.e. equivalent to the punctual category).
Let $\webleft (\mathsf{Sets},\otimes _{\mathsf{Sets}},\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}},\mathbb {1}_{\mathsf{Sets}},\lambda ',\rho ',\sigma '\webright )$ be a closed symmetric monoidal category satisfying Item 1 and Item 2. We need to show that the identity functor
for the component of this isomorphism at $\webleft (A,B\webright )$.
Constructing an Isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$
Since $\otimes _{\mathsf{Sets}}$ is adjoint in each variable to $\webleft [-_{1},-_{2}\webright ]_{\mathsf{Sets}}$ by assumption and $\times $ is adjoint in each variable to $\mathsf{Sets}\webleft (-_{1},-_{2}\webright )$ by Chapter 2: Constructions With Sets, Item 2 of Proposition 2.3.5.1.2, uniqueness of adjoints () gives us natural isomorphisms
for the component of this isomorphism at $\webleft (A,B\webright )$.
Alternative Construction of an Isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$
Alternatively, we may construct a natural isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$ as follows:
Let $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Since $\otimes _{\mathsf{Sets}}$ is part of a closed monoidal structure, it preserves colimits in each variable by .
Since $A\cong \mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}_{a\in A}\text{pt}$ and $\otimes _{\mathsf{Sets}}$ preserves colimits in each variable, we have
naturally in $B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, where we have used that $\text{pt}$ is the monoidal unit for $\otimes _{\mathsf{Sets}}$. Thus $A\otimes _{\mathsf{Sets}}-\cong A\times -$ for each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Similarly, $-\otimes _{\mathsf{Sets}}B\cong -\times B$ for each $B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
By , we then have $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$.
Below, we’ll show that if a natural isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$ exists, then it must be unique. This will show that the isomorphism constructed above is equal to the isomorphism $\text{id}^{\otimes }_{\mathsf{Sets}|A,B}\colon A\otimes _{\mathsf{Sets}}B\to A\times B$ from before.
Constructing an Isomorphism $\text{id}^{\otimes }_{\mathbb {1}}\colon \mathbb {1}_{\mathsf{Sets}}\to \text{pt}$
We define an isomorphism $\text{id}^{\otimes }_{\mathbb {1}}\colon \mathbb {1}_{\mathsf{Sets}}\to \text{pt}$ as the composition
Monoidal Right Unity of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$
We can use the same argument we used to prove the monoidal left unity of the isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$ above. For completeness, we repeat it below.
Monoidality of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$
We have to show that the diagram
commutes. First, note that the diagram
commutes by the terminality of $\text{pt}$ (Chapter 2: Constructions With Sets, Construction 2.1.1.1.2). Since the map $!_{\text{pt}\times \webleft (\text{pt}\times \text{pt}\webright )}\colon \text{pt}\times \webleft (\text{pt}\times \text{pt}\webright )\to \text{pt}$ is an isomorphism (e.g. having inverse $\lambda ^{\mathsf{Sets},-1}_{\text{pt}}\circ \lambda ^{\mathsf{Sets},-1}_{\text{pt}}$), it follows that the diagram
also commutes. Taking inverses, we see that the diagram
commutes as well. Now, let $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, let $a\in A$, and consider the diagram
which we partition into subdiagrams as follows: Since:
Subdiagram (1) commutes by the naturality of $\alpha ^{\mathsf{Sets},-1}$.
Subdiagram (2) commutes by the naturality of $\text{id}^{\otimes ,-1}_{\mathsf{Sets}}$.
Subdiagram (3) commutes by the naturality of $\text{id}^{\otimes ,-1}_{\mathsf{Sets}}$.
Subdiagram ($\dagger $) commutes, as proved above.
Subdiagram (4) commutes by the naturality of $\text{id}^{\otimes ,-1}_{\mathsf{Sets}}$.
Subdiagram (5) commutes by the naturality of $\text{id}^{\otimes ,-1}_{\mathsf{Sets}}$.
Subdiagram (6) commutes by the naturality of $\alpha ^{\prime ,-1}$.
Braidedness of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$
We have to show that the diagram
commutes. First, note that the diagram
commutes by the terminality of $\text{pt}$ (Chapter 2: Constructions With Sets, Construction 2.1.1.1.2). Since the map $!_{\text{pt}\times \text{pt}}\colon \text{pt}\times \text{pt}\to \text{pt}$ is invertible (e.g. with inverse $\lambda ^{\mathsf{Sets},-1}_{\text{pt}}$), the diagram
also commutes. Taking inverses, we see that the diagram
commutes as well. Now, let $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, let $a\in A$, and consider the diagram
which we partition into subdiagrams as follows:
Since:
Subdiagram (2) commutes by the naturality of $\sigma ^{\mathsf{Sets},-1}$.
Subdiagram (5) commutes by the naturality of $\text{id}^{\otimes ,-1}$.
Subdiagram ($\dagger $) commutes, as proved above.
Subdiagram (4) commutes by the naturality of $\sigma ^{\prime ,-1}$.
Subdiagram (1) commutes by the naturality of $\text{id}^{\otimes ,-1}$.
Uniqueness of the Isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$
Let $\phi ,\psi \colon -_{1}\otimes _{\mathsf{Sets}}-_{2}\Rightarrow -_{1}\times -_{2}$ be natural isomorphisms. Since these isomorphisms are compatible with the unitors of $\mathsf{Sets}$ with respect to $\times $ and $\otimes $ (as shown above), we have
for each $B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$. Now, let $a\in A$ and consider the naturality diagrams
for $\phi $ and $\psi $ with respect to the morphisms $\webleft [a\webright ]$ and $\text{id}_{B}$. Having shown that $\phi _{\text{pt},B}=\psi _{\text{pt},B}$, we have
for each $\webleft (a,b\webright )\in A\times B$. Therefore we have
\[ \phi _{A,B}=\psi _{A,B} \]
for each $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and thus $\phi =\psi $, showing the isomorphism $\mathord {\otimes _{\mathsf{Sets}}}\cong \mathord {\times }$ to be unique.