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Not necessarily forgetful, but otherwise yes: functors between categories of models for algebraic theories automatically satisfy the solution set condition. (It is also equivalent to asking when such functors preserve finite coproducts, as they automatically preserve sifted colimits.) However, this property doesn't seem practical to check in concrete examples, which is why I am interested in a syntactic characterisation.
I must admit I do not know of any examples beyond the one-object case! The condition seems rather strong, which suggests to me that there probably aren't many natural examples, but I believe there must be some (especially if Bénabou thought them interesting enough to study). I would hope that, if a reference exists, it also contains examples, but I would also be interested in an example even without a reference.
@Lolman: what I am asking you is why you think it is not possible. You ask "What fails?" which implies you have read somewhere that it doesn't work. Otherwise, I would have expected you to ask "Is there a 3-category of bicategories, lax functors, ICONs, and modifications?".
@Lolman: I think you misread my comment. I literally asked about the final paragraph of your question. The blog post is about pseudo/lax/colax transformations between lax functors, not ICONs.
Lack–Miranda's paper characterises the inclusion $\text{Mnd}(X) \to \text{EM}(X)$, so it is not helpful if you want to characterise $\text{Mnd}(X)$ in isolation.
