The skew associator of the left tensor product of pointed sets is the natural transformation
\[ \alpha ^{\mathsf{Sets}_{*},\lhd }\colon {\lhd }\circ {\webleft ({\lhd }\times \text{id}_{\mathsf{Sets}_{*}}\webright )}\Longrightarrow {\lhd }\circ {\webleft (\text{id}_{\mathsf{Sets}_{*}}\times {\lhd }\webright )}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathsf{Sets}_{*},\mathsf{Sets}_{*},\mathsf{Sets}_{*}}} \]
as in the diagram
whose component
\[ \alpha ^{\mathsf{Sets}_{*},\lhd }_{X,Y,Z} \colon \webleft (X\lhd Y\webright )\lhd Z \to X\lhd \webleft (Y\lhd Z\webright ) \]
at $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ is given by
\begin{align*} \webleft (X\lhd Y\webright )\lhd Z & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|Z|\odot \webleft (X\lhd Y\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|Z|\odot \webleft (|Y|\odot X\webright )\\ & \cong \bigvee _{z\in Z}|Y|\odot X\\ & \cong \bigvee _{z\in Z}\webleft (\bigvee _{y\in Y}X\webright )\\ & \to \bigvee _{\webleft [\webleft (z,y\webright )\webright ]\in \bigvee _{z\in Z}Y}X\\ & \cong \bigvee _{\webleft [\webleft (z,y\webright )\webright ]\in |Z|\odot Y}X\\ & \cong ||Z|\odot Y|\odot X\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|Y\lhd Z|\odot X\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X\lhd \webleft (Y\lhd Z\webright ),\end{align*}
where the map
\[ \bigvee _{z\in Z}\webleft (\bigvee _{y\in Y}X\webright )\to \bigvee _{\webleft (z,y\webright )\in \bigvee _{z\in Z}Y}X \]
is given by $\webleft [\webleft (z,\webleft [\webleft (y,x\webright )\webright ]\webright )\webright ]\mapsto \webleft [\webleft (\webleft [\webleft (z,y\webright )\webright ],x\webright )\webright ]$.
Firstly, note that, given $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, the map
\[ \alpha ^{\mathsf{Sets}_{*},\lhd }_{X,Y,Z} \colon \webleft (X\lhd Y\webright )\lhd Z \to X\lhd \webleft (Y\lhd Z\webright ) \]
is indeed a morphism of pointed sets, as we have
\[ \alpha ^{\mathsf{Sets}_{*},\lhd }_{X,Y,Z}\webleft (\webleft (x_{0}\lhd y_{0}\webright )\lhd z_{0}\webright )=x_{0}\lhd \webleft (y_{0}\lhd z_{0}\webright ). \]
Next, we claim that $\alpha ^{\mathsf{Sets}_{*},\lhd }$ is a natural transformation. We need to show that, given morphisms of pointed sets
\begin{align*} f & \colon \webleft (X,x_{0}\webright ) \to \webleft (X',x'_{0}\webright ),\\ g & \colon \webleft (Y,y_{0}\webright ) \to \webleft (Y',y'_{0}\webright ),\\ h & \colon \webleft (Z,z_{0}\webright ) \to \webleft (Z’,z’_{0}\webright ) \end{align*}
the diagram
commutes. Indeed, this diagram acts on elements as
and hence indeed commutes, showing $\alpha ^{\mathsf{Sets}_{*},\lhd }$ to be a natural transformation. This finishes the proof.
