The set of bilinear morphisms of pointed sets from $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$ to $\webleft (Z,z_{0}\webright )$ is the set $\smash {\textup{Hom}^{\otimes }_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright )}$ defined by
\[ \textup{Hom}^{\otimes }_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\in \textup{Hom}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )\ \middle |\ \text{$f$ is bilinear}\webright\} . \]
