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 $S$.