# Monadic over Set implies regular

This is a classic result. Here is one way of getting it:

From [AHS](http://katmat.math.uni-bremen.de/acc/acc.pdf):

- 20.21 Def: A monad is regular if it preserves regular epis.

- 20.22: Every monad on Set is regular.

- 20.28: Let U:A-->Set be monadic. Then (Surjections,Injections) is a factorisation system on A.

- 20.34 with 15.12, 15.13: Let U:A-->Set be monadic. Then (Surjections,Injections) = (RegEpi,Mono) in A.

Since the results in Chapter 20 of [AHS] are formulated in terms of factorisation systems for sources the following is worth noting.

- Every factorisation system $(E,\bf M)$ for sources induces a factorisation system $(E,M)$ just by restricting to sources that consists of one arrow only.
- Conversely, 15.21 says that if a category with a factorisation system $(E,M)$ has products and is E-cowellpowered and all arrows in $E$ are epi, then $(E,M)$ can be extended uniquely to a factorisation system for sources.

To summarise, for the kind of categories we are interested in, the distinction between factorisation system4 (for morphisms) and factorisation systems for sources can be ignored.