The diagonal of the left tensor product of pointed sets is the natural transformation
whose component
\[ \Delta ^{\lhd }_{X}\colon (X,x_{0})\to (X\lhd X,x_{0}\lhd x_{0}) \]
at $(X,x_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$ is given by
\[ \Delta ^{\lhd }_{X}(x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x\lhd x \]
for each $x\in X$.
Being a Morphism of Pointed Sets
We have
\[ \Delta ^{\lhd }_{X}(x_{0})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{0}\lhd x_{0}, \]
and thus $\Delta ^{\lhd }_{X}$ is a morphism of pointed sets.
Naturality
We need to show that, given a morphism of pointed sets
\[ f\colon (X,x_{0})\to (Y,y_{0}), \]
the diagram
commutes. Indeed, this diagram acts on elements as and hence indeed commutes, showing $\Delta ^{\lhd }$ to be natural.
