Say we have a Grothendieck fibration $p : E \to B$ and a monad $T$ on $B$ and a lift $T'$ of $T$ to $E$, i.e. a monad on $E$ such that $pT' = Tp$ and $p$ preserves $\eta, \mu$.


Then because the Eilenberg–Moore construction is functorial, we have a morphism $EM(p)$ from $T'\text{-}Alg$ to $T\text{-}Alg$. Is $EM(p)$ generally a fibration? If not, under what conditions is $EM(p)$ a fibration? 



Does the Eilenberg Moore Construction Preserve fibrations?
Let $C$ be a 2-category and $Mnd(C)$ the 2-category of monads in $C$. As explained by Street in [the formal theory of monads](https://doi.org/10.1016/0022-4049(72)90019-9), the Eilenberg-Moore construction is right 2-adjoint to the inclusion 2-functor $C \to Mnd(C)$ sending an object to the identity monad on it. Therefore it preserves 2-limits.


A [fibration in a 2-category](https://ncatlab.org/nlab/show/fibration+in+a+2-category) may be defined in terms of certain 2-limits (namely slice categories) and adjoints, and so is preserved by any 2-functor preserving 2-limits, and in particular by the Eilenberg-Moore construction.


Thus it's sufficient for $p$ to be a fibration object in the 2-category $Mnd(C)$.


