Can universal constructions be described as fixed points of functors on Cat? Suppose $Cat$ is the category of (small) categories and functors. My question is the following:
Can we characterize different categorical notions which are given by universal properties (i.e. "$\mathcal{C}$ has a terminal object", "$\mathcal{C}$ has binary products", "$\mathcal{C}$ is a Cartesian closed category", etc... by results such as: $$ \mathcal{C} \text{ has all binary products if and only if } \mathcal{F}(\mathcal{C}) \simeq \mathcal{C} $$ where $\mathcal{F} : Cat \to Cat$ is some (large) functor, thought of, in this case for instance, as a functor which takes $\mathcal{C}$ and "closes" it under binary products.
This seems like a natural generalization from e.x. the construction of algebraic closures in field theory to categories, but I have yet to see any results like this. Have people considered such characterizations of categorical notions in the literature? Is such a characterization possible always, never, or only in certain instances?
 A: A common situation is that we have an endofunctor $F$ on a category $C$ together with a natural map $\eta: 1\to F$, which could be dualized. Then we say an object $x$ is $F$-complete if $\eta_x$ is an isomorphism. For a reasonable notion of completion, we should assume $Fx$ is always complete. A natural way to enforce this is to assume that $\eta$ is the unit of a monad structure on $F$ whose multiplication is invertible; a completion is a kind of free construction, so we should expect a monad to appear. These extra assumptions make $F$ the monad induced by the inclusion of some reflective subcategory. 
Now, things like categories with binary products are certainly monadic over categories, but not via an idempotent monad as they form non-full subcategories. And you can't get out of this by taking categories with binary products and arbitrary functors, which do form a full subcategory: if there were a left adjoint to this inclusion, categories with binary products would be closed under all limits, which they are not (e.g. equalizers, or more dramatically, even flexible things like pseudo-equalizers, so there's not even a bicategorical left adjoint.) 
So these things can't be done using an idempotent monad. This situation seems a bit unusual to me from a classical algebra perspective: categories with binary products are a non-full subcategory, not containing all objects, containing all isomorphisms (or equivalences!) I'm failing to think of serious classical examples of this. 
Of course, this question doesn't have to be addressed via idempotent monads. Indeed, algebraic closure is not even an endofunctor of fields, let alone a monad. But such constructions are rare. 
