If $S$ and $T$ are monads on a category $C$, and $\lambda:S\to T$ is a morphism of monads, it is well-known that there is a functor $\lambda^*:C^T\to C^S$ which assigns to the $T$-algebra $(A,a:TA\to A)$ the $S$-algebra $(A,a\circ\lambda:SA\to A)$.
Does a similar phenomenon happen for pseudomonads and their pseudoalgebras (as defined for example here)?
A reference would be welcome too.
Yes, Theorem 3.4 of Gambino–Lobbia's On the formal theory of pseudomonads and pseudodistributive laws establishes that pseudomonad morphisms are in correspondence with liftings to pseudoalgebras. They work more generally in the setting of pseudomonads in a Gray-category, and with pseudomonads on different objects, rather than pseudomonads on a fixed object, but the theorem specialises to the result you are looking for.
