I would like to show that the category of sets and spans between them, seen as a $(2,1)$-category, is Cauchy complete, i.e. has splitting of (homotopicaly coherent) idempotent.
Ideally I would also like to prove that the retract of a set $X$ in this bi-category are all subsets of $X$, and that all this also works if we replace set by a category with finite limits.
I think I have a poof of this (If I'm not mistaken), but it is very long and technical and I wanted to know if there is literature about this, or if there is a simpler or more conceptual way to obtain these results.
