It is well known (e.g. Palmgren–Vickers's [Partial Horn logic and cartesian categories](http://www2.math.uu.se/~palmgren/partialalgebras_pre.pdf)) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / cartesian theories) may be characterised as small finitely complete categories. Here, the set of sorts is not fixed, and varies according to the category.

For a fixed set $S$, is there a known category theoretic characterisation of $S$-sorted essentially algebraic theories, so that (for example) one-sorted essentially algebraic theories may be characterised as certain finitely complete categories?

(Note that I am assuming that the equivalence between quasi-equational/partial Horn/essentially algebraic/cartesian theories continues to hold when the set of sorts is fixed. If this is not the case, I shall update my question accordingly.)

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There are some related results in the literature – I've listed the ones I know of below – but I have not found a characterisation for essentially algebraic theories.
- Keane proves in [Abstract Horn theories](https://link.springer.com/chapter/10.1007/BFb0061292) that a small category is equivalent to a one-sorted Horn theory iff it is finitely complete and has an object $I$ which is $\mathcal M$-injective and such that every object is a $\mathcal M$-subobject of $I^n$ for some $n \in \mathbb N$, for $\mathcal M$ a left-exact class of monomorphisms.
- Adámek and Porst prove in [Algebraic Theories of Quasivarieties](http://www.math.uni-bremen.de/~porst/dvis/AlgTheocorr.pdf) that a small category is equivalent to the theory of a one-sorted quasivariety iff it is finitely complete and has an object $I$ which is regular injective and such that every object is a regular subobject of $I^n$ for some $n \in \mathbb N$.
- $S$-sorted algebraic theories are well-known to be equivalent to strict finite product-preserving functors from the free category with finite products on $S$.

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I should note that a characterisation is deducible from the results of Di Liberti–Loregian–Nester–Sobociński's [Functorial Semantics for Partial Theories](https://arxiv.org/abs/2011.06644), as the finitely complete categories induced by (Cauchy complete) $S$-sorted partial algebraic theories; however, this is not a very direct characterisation and I am hoping for one along the lines of the previous examples.