The diagonal of the left tensor product of pointed sets is the natural transformation

whose component

\[ \Delta ^{\lhd }_{X}\colon \webleft (X,x_{0}\webright )\to \webleft (X\lhd X,x_{0}\lhd x_{0}\webright ) \]

at $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ is given by

\[ \Delta ^{\lhd }_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x\lhd x \]

for each $x\in X$.

Being a Morphism of Pointed Sets
We have

\[ \Delta ^{\lhd }_{X}\webleft (x_{0}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{0}\lhd x_{0}, \]

and thus $\Delta ^{\lhd }_{X}$ is a morphism of pointed sets.

Naturality
We need to show that, given a morphism of pointed sets

\[ f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ), \]

the diagram

commutes. Indeed, this diagram acts on elements as

and hence indeed commutes, showing $\Delta ^{\lhd }$ to be natural.


Noticed something off, or have any comments? Feel free to reach out!


You can also use the contact form below: