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.