For a category $C$, let $C-Set$ denote the category of set-valued functors $\delta\colon C\to Set$. Given categories $C$ and $D$, and a functor $F\colon C\to D$, composition with $F$ yields a functor that I'll denote by $$\Delta_F\colon D-Set\longrightarrow C-Set.$$ The functor $\Delta_F$ has both a left adjoint, which I'll denote by $\Sigma_F\colon C-Set\longrightarrow D-Set$, and a right adjoint $\Pi_F\colon C-Set\longrightarrow D-Set$. One then has a monad $M = \Delta_F\Sigma_F$ on $C-Set$ and a monad $N=\Pi_F\Delta_F$ on $D-Set$. Let $M-alg=(C-Set)^M$ denote the category of $M$-algebras on $C-Set$, and similarly, let $N-alg=(D-Set)^N$ denote the category of $N$-algebras on $D-Set$.

I'll say that $\Delta_F$ is monadic if the obvious functor $D-Set\longrightarrow M-alg$ is an equivalence of categories, and I'll say that $\Pi_F$ is monadic if the obvious functor $C-Set\longrightarrow N-alg$ is an equivalence of categories.

Questions:

1. Under what conditions on $F$ is $\Delta_F$ monadic?

2. Under what conditions on $F$ is $\Pi_F$ monadic?

Thanks!