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
\[ \alpha ^{\mathsf{Sets}_{*},\rhd }_{X,Y,Z} \colon X\rhd \webleft (Y\rhd Z\webright ) \to \webleft (X\rhd Y\webright )\rhd Z \]
is indeed a morphism of pointed sets, as we have
\[ \alpha ^{\mathsf{Sets}_{*},\rhd }_{X,Y,Z}\webleft (x_{0}\rhd \webleft (y_{0}\rhd z_{0}\webright )\webright )=\webleft (x_{0}\rhd y_{0}\webright )\rhd z_{0}. \]
Next, we claim that $\alpha ^{\mathsf{Sets}_{*},\rhd }$ 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 $\alpha ^{\mathsf{Sets}_{*},\rhd }$ to be a natural transformation. This finishes the proof.
