Term models for bounded infinitary equational theories

Question

In this answer https://mathoverflow.net/a/335864/64128 to a question about whether it is possible in general to construct initial term models for infinitary equational theories, it is remarked that if all the arities of the operations are choice sets (and the operations themselves constitute a set), then the initial term model for such a theory can be constructed. So does this hold when the arities of the operations are all bounded by some regular cardinal?

Answer 1

No. Indeed, this is answered by Andreas Blass's paper Words, free algebras, and coequalizers mentioned in the other answer to the question you linked. Namely, in section 9 he gives an example of an equational theory for which all operations have countable arity such that in a certain model of ZF, there does not exist any initial algebra. (Specifically, the theory has a constant $0$, a unary "successor" operation, and a $\omega$-ary "supremum" operation, together with axioms that say these behave reasonably. In a model where every limit ordinal has countable cofinality, you can reach all ordinals by iterating these operations, and this means that the initial algebra could not be a set.)