I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras over some monad or something to do with enriched categories but I can't find anything.
Specifically, I would like something more "formal" than just "categories equipped with necessary adjunctions and functors that preserve them on the nose".
Even better would be if this definition could be very easily related to definitions of related categories like the category of cartesian categories or monoidal categories.