Pseudoalgebras and Non-canonical Isomorphisms

Given a pseudomonad T , we prove that a lax T -morphism between pseudoalgebras is a T -pseudomorphism if and only if there is a suitable (possibly non-canonical) invertible T -transformation. This result encompasses several results on non-canonical isomorphisms , including Lack’s result on normal monoidal functors between braided monoidal categories, since it is applicable in any 2-category of pseudoalgebras, such as the 2-categories of monoidal categories, cocomplete categories, bicategories, pseudofunctors and so on.