Characterisation of presentations for varietal large equational theories Let $T : \mathbf{Set}^\mathrm{op} \to \mathscr T$ be a large equational theory (i.e. a bijective-on-objects product-preserving functor). Following Linton in Some Aspects of Equational Categories, we call $T$ varietal  if $\mathscr T$ is locally small. A large equational theory may be presented by a class $\Omega$ of set-indexed (i.e. possibly infinitary) operations and a class $E$ of equations.
May the presentations of varietal large equational theories be neatly characterised? If not, is there a fundamental obstacle to doing so?
In the case that there is not a precise characterisation, I would also be interested in necessary, and sufficient, conditions for a large presentations to be varietal. One sufficient condition is that $\Omega$ be bounded in that there exists some cardinal $\kappa$ such that every operation has arity less than $\kappa$, but I would like to know if there are stronger conditions: for instance, one that implies the theory of sup-lattices, or of compact Hausdorff spaces, is varietal.
 A: The following are equivalent:

*

*$\mathcal T$ is varietal


*Each free algebra $T \in \mathcal T$ is small


*For each arity, there are a small number of $E$-equivalence classes of words in the language $\Omega$.
$(1) \Rightarrow (2)$ holds because the underlying set of an algebra can be identified with morphisms from the free algebra on 1 generator. $(2) \Rightarrow (1)$ holds because there are a small set of functions between any two small sets. The equivalence between $(2)$ and $(3)$ is hopefully clear.
I think there may not be much more to say at this level of generality:  In cases where condition (3) is hard to check directly, then I suspect one might need to use more information about how $(\Omega, E)$ is presented in order to get a more usable criterion.
For instance, if $(\Omega, E)$ gives rise to a varietal theory, then it must be equivalently axiomatizable in the form $(\Omega', E')$ where $\Omega'$ has a small number of operations of each arity. But this condition is not sufficient.
