Recall the following theorem of Linton:


A functor $U:E\to \operatorname{Set}$ is monadic if 


1. $U$ has a left adjoint,
2. $E$ admits kernel pairs and coequalizers,
3. A parallel pair $R \rightrightarrows S$ in $E$ is a kernel pair if and only if its image under $U$ is so, and
4. A morphism $A\to B$ in $E$ is a regular epimorphism if and only if its image under $U$ is so.


According to the [nLab page](http://ncatlab.org/nlab/show/monadicity+theorem#monadicity_over_set_15), this can be modified such that we may replace the category $\operatorname{Set}$ in the above with the category $\operatorname{Set}^C$ for any small category $C$. However, I'm having a bit of a time finding a place that proves this.


I'd be extremely happy with a reference and will accept any answer with such a reference.



The (co)monadicity theorem relative to a presheaf topos
The result you are after is Theorem 1.2 of "A Monadicity Theorem" by Borceux and Day, published in the Bulletin of the Australian Mathematical Society in 1977, and the paper is available online. If you have access to Mathscinet reviews but not the paper you can even see the result stated in detail in the review by Kelly.


Their Theorem 1.2 is exactly as you state it above for E over Set with the exception that Set is replaced by an arbitrary category with kernel pairs and coequalisers, and so can be any presheaf category.


