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!



When do functors induce monadic adjunctions to presheaf categories
I will try to give an answer to your first question.


The functor $\Delta\_F$ verifies automatically nearly all the conditions of the monadicity theorem:


* it is a right adjoint;
* it is a left adjoint with cocomplete domain, and thus coequalizers exist in the source and are preserved by $\Delta\_F$.


It remains to see when the functor $\Delta\_F$ is conservative, i.e. reflects isomorphisms. This is true, for example, in each of the following two cases:


* $F$ is essentially surjective;
* $\Delta\_F$ is full and faithful. This condition holds if $F$ induces an equivalence of Cauchy completions, i.e. $F$ is full and faithful and every object in the codomain is a retract of an object in the image. $\Delta\_F$ is also full and faithful when $F$ is a reflection onto a reflective subcategory, or more generally when $F$ formally inverts some arrows (such as the map from a model category to its homotopy category).


