Category of multisorted Lawvere's theories

Question

Multisorted Lawvere's theory consists of

• sets of sorts $S$
• small category $T$
• a preserving-product essentially bijective functor $(\mathrm{finSet}/S)^{\mathrm{op}} \to T$

How is category of multisorted Lawvere's theories correctly defined? I couldn't find it in the literature (the category of one-sort Lawvere's theories is defined, for example, in Martin Hyland, John Power, The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads, see p. 4-5). The first thing that comes to mind is a morphism of sets of sorts $S_1 \to S_2$ and a product-preserving functor $T_1 \to T_2$ so that the corresponding square diagram (involving $(\mathrm{finSet}^{S_i})^{\mathrm{op}}$) is commutative. Is this the correct definition?

Answer 1

A definition can be found in 4.1 (page 8) of Charles Rezk's paper "Every Homotopy Theory of Simplicial Algebras Admits a Proper Model." The category of $J$-sorted theory is the category of monoids in $\mathcal{S}^{f\mathcal{S}(J)}$, with a product Rezk defines in 3.8.

In Rezk's notation, $\mathcal{S}$ is the category of sets and $f\mathcal{S}$ is a fixed skeleton of the category of finite sets. The category $\mathcal{S}^{f\mathcal{S}}$ is the category of functors from $f\mathcal{S}$ to $\mathcal{S}$.

Fix a set $J$. The category $f\mathcal{S}(J,J)$ is defined above 3.8 and is equivalent to the category of functors from finite $J$-graded sets to $J$-graded sets. Composition gives the monoidal product. This is completely analogous to the situation of $J$-colored operads. Just as you can assemble all the categories of $J$-colored operads into a category of pairs $(J,O)$ where $J$ is a set and $O$ is a $J$-colored operad (e.g., this is a Grothendieck construction), so too can you assemble the categories of $J$-sorted theories into a category of multisorted theories. The morphisms are the same as in any Grothendieck construction.