In "Partial Horn logic and cartesian categories", E. Palmgren and S. J. Vickers state without proof that "The theory of categories is not algebraic." Is there a reference, or an elementary argument, for this fact? In particular, I'm interested in the setting where "algebraic" refers to the multisorted, potentially infinitary setting.


To be precise, this would entail showing that there is no monadic functor from $\mathbf{Cat} \to \mathbf{Set}/S$ for any set $S$.



Why is the theory of small categories not algebraic?
This follows from two Facts:


**1)** A category monadic over Set/S is always an [exact category](https://ncatlab.org/nlab/show/exact+category). That is it has quotient by equivalence relation that are effective and universal. It is in particular a [regular category](https://ncatlab.org/nlab/show/regular+category). This is showed for example [here.](https://ncatlab.org/nlab/show/colimits+in+categories+of+algebras#exact)


**2)** The category of categories is not a regular category. An explicit example of regularity (and hence exactness) failing in the category of categories (or posets, or topological spaces) can be found in the [example section](https://ncatlab.org/nlab/show/regular+category#examples) of the nLab page.


