In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint.
Now how does this generalize for the relative case? Let $C$, $D$ be topoi, let functor $T : C \rightarrow D$ be a monad relative to $J : C \to D$ and let $EM(T)$ be its Eilenberg-Moore category as defined in "Monads need not be endofunctors". Do we have any conditions for when this category is a topos? Such as if $T$ has a right adjoint or more generally a suitable relative right adjoint.
Notably normally the proof for Eilenberg-Moore category goes through the co-Eilenberg-Moore category of left exact comonad (the right adjoint). But in the relative case the theory bifurcates into left-adjoint/monad and right-adjoint/comonad theory. Can we still translate a relative monad structure into a relative comonad structure along a relative adjunction under some conditions? Or perhaps replay the topos proof but unwind it?
