The left unitor of the smash product of pointed sets is the natural isomorphism
whose component
\[ \lambda ^{\mathsf{Sets}_{*}}_{X} \colon S^{0}\wedge X \xrightarrow {\cong }X \]
at $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ is given by
\begin{align*} 0\wedge x & \mapsto x_{0},\\ 1\wedge x & \mapsto x. \end{align*}
Well-Definedness
Let $\webleft [\webleft (x,y\webright )\webright ]=\webleft [\webleft (x',y'\webright )\webright ]$ be an element in $S^{0}\wedge X$. Then either:
-
We have $x=x'$ and $y=y'$.
-
Both of the following conditions are satisfied:
-
We have $x=0$ or $y=x_{0}$.
-
We have $x'=0$ or $y'=x_{0}$.
In the first case, $\lambda ^{\mathsf{Sets}_{*}}_{X}$ clearly sends both elements to the same element in $X$. Meanwhile, in the latter case both elements are equal to the basepoint $0\wedge x_{0}$ of $S^{0}\wedge X$, which gets sent to the basepoint $x_{0}$ of $X$.
Being a Morphism of Pointed Sets
As just mentioned, we have
\[ \lambda ^{\mathsf{Sets}_{*}}_{X}\webleft (0\wedge x_{0}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{0}, \]
and thus $\lambda ^{\mathsf{Sets}_{*}}_{X}$ is a morphism of pointed sets.
Invertibility
The inverse of $\lambda ^{\mathsf{Sets}_{*}}_{X}$ is the morphism
\[ \lambda ^{\mathsf{Sets}_{*},-1}_{X}\colon X\xrightarrow {\cong }S^{0}\wedge X \]
defined by
\[ \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}1\wedge x \]
for each $x\in X$. Indeed:
- Invertibility I. We have
\begin{align*} \webleft [\lambda ^{\mathsf{Sets}_{*},-1}_{X}\circ \lambda ^{\mathsf{Sets}_{*}}_{X}\webright ]\webleft (0\wedge x\webright ) & = \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webleft (\lambda ^{\mathsf{Sets}_{*}}_{X}\webleft (0\wedge x\webright )\webright )\\ & = \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webleft (x_{0}\webright )\\ & = 1\wedge x_{0}\\ & = 0\wedge x, \end{align*}
and
\begin{align*} \webleft [\lambda ^{\mathsf{Sets}_{*},-1}_{X}\circ \lambda ^{\mathsf{Sets}_{*}}_{X}\webright ]\webleft (1\wedge x\webright ) & = \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webleft (\lambda ^{\mathsf{Sets}_{*}}_{X}\webleft (1\wedge x\webright )\webright )\\ & = \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webleft (x\webright )\\ & = 1\wedge x \end{align*}
for each $x\in X$, and thus we have
\[ \lambda ^{\mathsf{Sets}_{*},-1}_{X}\circ \lambda ^{\mathsf{Sets}_{*}}_{X}=\text{id}_{S^{0}\wedge X}. \]
- Invertibility II. We have
\begin{align*} \webleft [\lambda ^{\mathsf{Sets}_{*}}_{X}\circ \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webright ]\webleft (x\webright ) & = \lambda ^{\mathsf{Sets}_{*}}_{X}\webleft (\lambda ^{\mathsf{Sets}_{*},-1}_{X}\webleft (x\webright )\webright )\\ & = \lambda ^{\mathsf{Sets}_{*},-1}_{X}\webleft (1\wedge x\webright )\\ & = x \end{align*}
for each $x\in X$, and thus we have
\[ \lambda ^{\mathsf{Sets}_{*}}_{X}\circ \lambda ^{\mathsf{Sets}_{*},-1}_{X}=\text{id}_{X}. \]
This shows $\lambda ^{\mathsf{Sets}_{*}}_{X}$ to be invertible.
Naturality
We need to show that, given a morphism of pointed sets
\[ f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ), \]
the diagram
commutes. Indeed, this diagram acts on elements as
and
and hence indeed commutes, showing $\lambda ^{\mathsf{Sets}_{*}}$ to be a natural transformation.
Being a Natural Isomorphism
Since $\lambda ^{\mathsf{Sets}_{*}}$ is natural and $\lambda ^{\mathsf{Sets}_{*},-1}$ is a componentwise inverse to $\lambda ^{\mathsf{Sets}_{*}}$, it follows from Chapter 8: Categories, Item 2 of Proposition 8.8.6.1.2 that $\lambda ^{\mathsf{Sets}_{*},-1}$ is also natural. Thus $\lambda ^{\mathsf{Sets}_{*}}$ is a natural isomorphism.
