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It is a classical result of categorical algebra, due to Lawvere and Linton,that finitary varieties of algebras (in the sense of Birkhoff) are duallyequivalent to finitary monads on $Set$. Recent work of Ad\'amek, Dost\'al, andVelebil has established that analogous results also hold in certain enrichedcontexts. Specifically, taking $V$ to be one of the cartesian closed categories$\mathsf{Pos}$, $\mathsf{UltMet}$, $\omega$-$\mathsf{CPO}$, or $\mathsf{DCPO}$of respectively posets, (extended) ultrametric spaces, $\omega$-cpos, or dcpos,Ad\'amek, Dost\'al, and Velebil have shown that a suitable category of$V$-enriched varieties of algebras is dually equivalent to the category ofstrongly finitary $V$-monads on $V$.  In this paper, we extend and generalize these results in two ways: byallowing $V$ to be an arbitrary complete and cocomplete cartesian closedcategory that is concrete over $Set$, and by also considering the multi-sortedcase. Given a set $S$ of sorts, we define a suitable notion of (finitary)$V$-enriched $S$-sorted variety, and we say that a $V$-monad on the product$V$-category $V^S$ is strongly finitary if its underlying $V$-endofunctor isthe left Kan extension of its restriction to a suitable full sub-$V$-categoryof $V^S$. Our main result is that the category of $V$-enriched $S$-sortedvarieties is dually equivalent to the category of strongly finitary $V$-monadson $V^S$. By taking $S$ to be a singleton and $V$ to be $\mathsf{Pos}$,$\mathsf{UltMet}$, $\omega$-$\mathsf{CPO}$, or $\mathsf{DCPO}$, we thus recoverthe aforementioned results of Ad\'amek, Dost\'al, and Velebil. We provideseveral classes of examples of $V$-enriched $S$-sorted varieties, many of whichadmit very concrete, syntactic formulations.

Strongly finitary monads and multi-sorted varieties enriched in cartesian closed concrete categories