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](https://www.sciencedirect.com/science/article/pii/S0001870899918819))?


A reference would be welcome too.



Morphism of pseudomonads induces pullback functors between pseudoalgebras
Yes, Theorem 3.4 of Gambino–Lobbia's [On the formal theory of pseudomonads and pseudodistributive laws](https://arxiv.org/abs/0907.1359) 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.


