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