Exercise 5.5.iii in Riehl's Category Theory in Context consists in proving the following result:
Proposition 5.5.8 (RTT). If $U: \mathsf{D} \rightarrow \mathsf{C}$ has a left adjoint and if
- $\mathsf{D}$ has coequalizers of reflexive¹ pairs,
- $U$ preserves coequalizers of reflexive pairs, and
- $U$ reflects isomorphisms, then $U$ is monadic.
¹ A parallel pair $f,g:A\rightrightarrows B$ is reflexive if both maps admit a common section $s:B\to A$.
I tried to show that these conditions imply the precise tripleability condition in Theorem 5.5.1 from Riehl's text, i.e., that conditions 1, 2, 3 of Proposition 5.5.8 imply that $U$ creates coequalizers of $U$-split pairs, but I have been unable to achieve this. How should one proceed?
Okay, so it turns out it suffices to check 1+2+3 imply that $U$ creates coequalizers of reflexive $U$-split pairs [ref]. On the one hand, 1+2 imply that $U$ has all coequalizers for all reflexive $U$-split pairs, and that it preserves them. It is left to show that $U$ reflects coequalizers of reflexive $U$-split pairs. But $U$ is conservative, so we are done by application of this result.