Let $j: A \to B$ be a fully faithful functor.

When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$ and thus will be idempotent, because $A$ is reflective in $B$.

Rem 1. Let $j$ be a faithful functor with a left adjoint. Then the codensity monad $\text{Ran}_jj$ is idempotent.

I was wondering if this is still true removing the assumption of having a left adjoint.

In a relatively trivial way, one can reformulate what I said in the following way.

Rem 2. Let $j$ be a fully faithful functor. TFAE:

• $\text{Ran}_jj$ preserve itself.
• $\text{Ran}_jj$ is idempotent.

So, the question is finally the following,

Q1. Let $j$ be a faithful functor, is it true that one of the two equivalent conditions is verified? You can still assume that $j$ preserve all limits.

Q2. If not, are there some natural assumptions that will make it true?