Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto X\wedge Y$ define functors
\begin{gather*} \begin{aligned} X\wedge - & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ -\wedge Y & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ \end{aligned}\\ -_{1}\wedge -_{2} \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*}. \end{gather*}
In particular, given pointed maps
\begin{align*} f & \colon \webleft (X,x_{0}\webright ) \to \webleft (A,a_{0}\webright ),\\ g & \colon \webleft (Y,y_{0}\webright ) \to \webleft (B,b_{0}\webright ), \end{align*}
the induced map
\[ f\wedge g\colon X\wedge Y\to A\wedge B \]
is given by
\[ \webleft [f\wedge g\webright ]\webleft (x\wedge y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (x\webright )\wedge g\webleft (y\webright ) \]
for each $x\wedge y\in X\wedge Y$.