Let $T$ be a 2-monad on a nice 2-category $\mathcal K$, so that the inclusion $T\text{-}\mathbf{Alg}_s \to T\text{-}\mathbf{Alg}_l$ of the 2-category of (strict) $T$-algebras and strict $T$-algebra morphisms into the 2-category of $T$-algebras and lax $T$-algebra morphisms admits a left adjoint $({-})' : T\text{-}\mathbf{Alg}_l \to T\text{-}\mathbf{Alg}_s$, the lax morphism classifier. This adjunction induces a lax-idempotent 2-comonad $Q_l$ on $T\text{-}\mathbf{Alg}_s$ (this follows from Lemma 2.5 of Lack–Shulman's Enhanced 2-categories and limits for lax morphisms).
It is alluded, for instance, in this n-Category Café comment, that $Q_l$ is the "lax-idempotentification" of the free–forgetful 2-adjunction $T\text{-}\mathbf{Alg}_s \rightleftarrows \mathcal K$, in the sense of a 2-dimensional analogue of the transfinite construction of an idempotent monad from an arbitrary monad.
Is there a reference for this in the literature? If not, does it follow easily from existing results in the literature? Or is it categorical folklore?
(I would also be satisfied to receive a reference for the pseudo rather than the lax case.)
