Unwinding the notation for elements, we have
\begin{align*} \webleft [\webleft (x,\webleft [\webleft (y,z\webright )\webright ]\webright )\webright ] & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (x,y\rhd z\webright )\webright ]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x\rhd \webleft (y\rhd z\webright ) \end{align*}
and
\begin{align*} \webleft [\webleft (\webleft [\webleft (x,y\webright )\webright ],z\webright )\webright ] & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (x\rhd y,z\webright )\webright ]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (x\rhd y\webright )\rhd z. \end{align*}
So, in other words, $\alpha ^{\mathsf{Sets}_{*},\rhd }_{X,Y,Z}$ acts on elements via
\[ \alpha ^{\mathsf{Sets}_{*},\rhd }_{X,Y,Z}\webleft (x\rhd \webleft (y\rhd z\webright )\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (x\rhd y\webright )\rhd z \]
for each $x\rhd \webleft (y\rhd z\webright )\in X\rhd \webleft (Y\rhd Z\webright )$.
