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, 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 $C$.

Do we need to replace the kernel pairs and coequalizers with their appropriate $C$-indexed analogues?

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

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, 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.