The skew associator of the right tensor product of pointed sets is the natural transformation
\[ \alpha ^{\mathsf{Sets}_{*},\rhd } \colon {\rhd }\circ {(\operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}}\times {\rhd })} \Longrightarrow {\rhd }\circ {({\rhd }\times \operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}})}\circ {\mathbf{\alpha }^{\mathsf{Cats},-1}_{\mathsf{Sets}_{*},\mathsf{Sets}_{*},\mathsf{Sets}_{*}}} \]
as in the diagram
\[ \alpha ^{\mathsf{Sets}_{*},\rhd }_{X,Y,Z} \colon X\rhd (Y\rhd Z) \to (X\rhd Y)\rhd Z \]
at $(X,x_{0}),(Y,y_{0}),(Z,z_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$ is given by
\begin{align*} X\rhd (Y\rhd Z) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|X|\odot (Y\rhd Z)\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|X|\odot (|Y|\odot Z)\\ & \cong \bigvee _{x\in X}(|Y|\odot Z)\\ & \cong \bigvee _{x\in X}(\bigvee _{y\in Y}Z)\\ & \to \bigvee _{[(x,y)]\in \bigvee _{x\in X}Y}Z\\ & \cong \bigvee _{[(x,y)]\in |X|\odot Y}Z\\ & \cong ||X|\odot Y|\odot Z\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|X\rhd Y|\odot Z\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(X\rhd Y)\rhd Z,\end{align*}
where the map
\[ \bigvee _{x\in X}(\bigvee _{y\in Y}Z)\to \bigvee _{[(x,y)]\in \bigvee _{x\in X}Y}Z \]
is given by $[(x,[(y,z)])]\mapsto [([(x,y)],z)]$.
