Filtered colimits and the free frame construction

Question

Let $F : C \to D$ be a functor between categories with filtered colimits, and let Cat be the category of categories. Let $(M, μ, η)$ be a monad on Cat. What properties of $M$ are necessary to ensure that

[p] F preserves filtered colimits if and only if M F has a right adjoint.

Here's a slightly different situation that I'm interested in:

Let

• $\text{Poset}$ be the category of partially ordered sets (posets).
• $\text{CompPoset}$ be the category of complete partial orders with join preserving poset maps as maps
• $\text{Meet}$ be the category of partial orders with meets, and with meet preserving poset maps as maps.
• $\text{Frame}$ be the category of frames. Maps of frames must preserve meets and joins.
• use op for the opposite poset, opposite complete poset, opposite map, etc.

There are adjunctions

• $L : \text{Poset} \leftrightarrow \text{Meet} : R$ ($L$ freely adds in meets and $R$ forgets)
• $CL : \text{CompPoset} \leftrightarrow \text{Frame} : CR$ ($CL$ freely adds in meets among complete partial orders and $CR$ forgets).

I am trying to prove or disprove:

1. For a complete poset $X$, $L X = CL X$. That is, $L X$ has the universal property of $CL X$.
2. For $f : X \rightarrow Y$ a map of partially ordered sets with filtered colimits, $L op f$ has a right adjoint if and only if $f$ preserves filtered colimits.
3. For $f: X \rightarrow Y$ a map of complete partially ordered sets, $f$ preserves filtered colimits if and only if $CL f$ has a right adjoint.

2 is the more important of these for me. And note that 3 follows from 1 and 2.