The diagonal of the smash product of pointed sets is the natural transformation
whose component
\[ \Delta ^{\wedge }_{X}\colon \webleft (X,x_{0}\webright )\to \webleft (X\wedge X,x_{0}\wedge x_{0}\webright ) \]
at $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ is given by the composition
in $\mathsf{Sets}_{*}$, and thus by
\[ \Delta ^{\wedge }_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x\wedge x \]
for each $x\in X$.
Being a Morphism of Pointed Sets
We have
\[ \Delta ^{\wedge }_{X}\webleft (x_{0}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{0}\wedge x_{0}, \]
and thus $\Delta ^{\wedge }_{X}$ is a morphism of pointed sets.
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 hence indeed commutes, showing $\Delta ^{\wedge }$ to be natural.
