nLab
tensor product theory

If $A$ and $B$ are algebraic theories, the algebraic theory $A\otimes B$ is characterized by the fact that its models can be identified with $A$-models in $B\mathrm{Mod}$, or equivalently as $B$-models in $A\mathrm{Mod}$. There are maps of theories $A\to A\otimes B$ and $B\to A\otimes B$ which are universal for maps of theories $A\to C$ and $B\to C$ whose images commute, for any theory $C$.