Let $\newcommand{\Fc}{\mathbf{Func}}\Fc$ be the category whose objects are functors between small categories and whose morphisms are commutative squares (the commutativity is given by a natural isomorphism, not equality). Let $\newcommand{\Mon}{\mathbf{Mon}}\Mon$ be the category of monads on small categories. Its objects are monads and the morphisms are monad morphisms. Let $S:C→C$ and $T:D→D$ be two monads. A monad morphism from $S$ to $T$ is a functor $F:C→D$ and a natural transformation $α : T∘F→F∘S$ respecting the units and the multiplications of the monads.
Then the category $\Mon$ embeds fully faithfully in the category $\Fc$ if we send $T:C→C$ to the forgetful functor $C^T→C$ of the Eilenberg-Moore category of the monad.
If a functor has a left adjoint, then it can be reflected in $\Mon$ (the composite of the left adjoint and the right adjoint is a monad). But is the converse true? If not, what is a counter-example? If it is true, what about the case of monads in an arbitrary 2-category?
Zhen Lin suggested to look at codensity monads, and it gives a negative answer to the following variation of this question. Fix a category $D$. Let $\newcommand{\Cat}{\mathbf{Cat}}\Cat_{/D}$ be the category of functors to $D$, with morphisms the triangles commuting up to natural isomorphism. Let $\Mon_D$ be the category of monads on $D$, with monad morphisms from $S$ to $T$ the natural transformations $T→S$ (yes, the order is reversed) commuting with the unit and multiplication. Then there is a fully faithful embedding $\Mon_D → \Cat_{/D}$ with essential image the monadic functors.
There are functors $F:C→D$ with no adjoint but which can be reflected in this full subcategory. If $F : C→D$ is a functor with a codensity monad $\newcommand{\Ran}{\mathrm{Ran}}\Ran_F(F)$, then $\Ran_F(F)$ is by definition the universal functor acting on $F$ on the right, but it is also the universal monad acting on $F$ on the right. (In exactly the same way, in a closed monoidal category, $X^X$ is the universal monoid acting on $X$.)
For instance, the functor $\mathbf{FinSet} → \mathbf{Set}$, which doesn't have any left adjoint, can be approximated by the monadic functor $\mathbf{KHaus} → \mathbf{Set}$.
But this universal property of the codensity monad doesn't transfer at all to the setting where we allow monad morphisms between different categories (or with non-identity functor $D→D$). For instance, take the empty inclusion $∅→\mathbf{Set}$. This generates the monadic functor $1 → \mathbf{Set}$ constant at $1$. Now, consider the functor $\mathbf{Set} → \mathbf{Set}$ sending every set to $∅$. This defines a morphism from $∅→\mathbf{Set}$ to $\mathbf{Grp} → \mathbf{Set}$, but there is no way to extend it to a morphism from $1→\mathbf{Set}$ to $\mathbf{Grp}→\mathbf{Set}$ since it would mean that we should find a group with an empty underlying set. (On the other hand, if $\mathbf{Set} → \mathbf{Set}$ is the identity instead of the functor sending everything to $0$, then there is a unique extension, given by the unique algebra with carrier $1$.)