# Why is the theory of small categories not algebraic?

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 $$S$$.

• One way to see it is that the category of categories is not a regular category, while model of algebraic theory (even multisorted and infinitary) are regular, even exact categories. – Simon Henry Mar 14 at 20:58