Let $F : A \to B$ be a functor, and suppose that the right Kan extension $T = Ran_F F : B \to B$ exists. Then $T$ is a monad, the codensity monad of $F$. Moreover, unless I'm mistaken there is a canonical factorization of $F$ through the category of algebras of $T$: $A \xrightarrow{\tilde F} Alg(T) \xrightarrow{U^T} B$ (at least up to some canonical 2-cell?).

I want to think of $\tilde F : B \to Alg(T)$ as a "jazzed-up" version of $F$, which remembers everything that $F$ remembers in a "maximally structured" way. I'd like to express this as some kind of universal property of $\tilde F$.

Question 1: Is the factorization $F = U^T \circ \tilde F$ part of some kind of factorization system whose right half is the monadic functors? If so, what is the left half (i.e. what are the characteristic properties of the functor $\tilde F$)?

Question 2: If we don't have a factorization system, then is there still something to be said about the passage $F \mapsto \tilde F$? Is it at least left adjoint to something (as a functor from $Cat_{A/}$ to some subcategory of $Cat_{A/}$, perhaps)?

If it helps to assume that $A,B,F$ have nice properties, that's fine by me. I'm also interested to understand the dual situation of the the density comonad.