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 $B Mod$, or equivalently as $B$-models in $A Mod$. 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 $C$.

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