The skew associator of the left tensor product of pointed sets is the natural transformation
\[ \alpha ^{\mathsf{Sets}_{*},\lhd }\colon {\lhd }\circ {({\lhd }\times \operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}})}\Longrightarrow {\lhd }\circ {(\operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}}\times {\lhd })}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathsf{Sets}_{*},\mathsf{Sets}_{*},\mathsf{Sets}_{*}}} \]
as in the diagram
\[ \alpha ^{\mathsf{Sets}_{*},\lhd }_{X,Y,Z} \colon (X\lhd Y)\lhd Z \to X\lhd (Y\lhd Z) \]
at $(X,x_{0}),(Y,y_{0}),(Z,z_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$ is given by
\begin{align*} (X\lhd Y)\lhd Z & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|Z|\odot (X\lhd Y)\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|Z|\odot (|Y|\odot X)\\ & \cong \bigvee _{z\in Z}|Y|\odot X\\ & \cong \bigvee _{z\in Z}(\bigvee _{y\in Y}X)\\ & \to \bigvee _{[(z,y)]\in \bigvee _{z\in Z}Y}X\\ & \cong \bigvee _{[(z,y)]\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 (Y\lhd Z),\end{align*}
where the map
\[ \bigvee _{z\in Z}(\bigvee _{y\in Y}X)\to \bigvee _{(z,y)\in \bigvee _{z\in Z}Y}X \]
is given by $[(z,[(y,x)])]\mapsto [([(z,y)],x)]$.
