As shown in Item 1 of Proposition 2.1.3.1.3, the Cartesian product of sets defines a functor
\[ -_{1}\times -_{2}\colon \mathsf{Sets}\times \mathsf{Sets}\to \mathsf{Sets}. \]
This functor is the $\webleft (k,\ell \webright )=\webleft (-1,-1\webright )$ case of a family of functors
\[ \otimes _{k,\ell }\colon \mathsf{Mon}_{\mathbb {E}_{k}}\webleft (\mathsf{Sets}\webright )\times \mathsf{Mon}_{\mathbb {E}_{\ell }}\webleft (\mathsf{Sets}\webright )\to \mathsf{Mon}_{\mathbb {E}_{k+\ell }}\webleft (\mathsf{Sets}\webright ) \]
of tensor products of $\mathbb {E}_{k}$-monoid objects on $\mathsf{Sets}$ with $\mathbb {E}_{\ell }$-monoid objects on $\mathsf{Sets}$; see 
.
