Pseudoalgebras and non-canonical isomorphisms

Given a pseudomonad $\mathcal{T}$, we prove that a lax $\mathcal{T}$-morphism between pseudoalgebras is a $\mathcal{T}$-pseudomorphism if and only if there is a suitable (possibly non-canonical) invertible $\mathcal{T}$-transformation. This result encompasses several results on $\mathit{\text{non-canonical}}\mathit{\text{isomorphisms}}$, 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, pseudofunctors and so on.