Recall that for a monad $(T,\eta,\mu)$ on a category $C$, the Kleisli category $C_T$ has as objects the objects of $C$ and as morphisms $C_T(x,y) = C(x,T y)$. A morphism $f\in C_T(x,y) = C(x,T y)$ is said to be thunkable if $T\eta \circ f = T f \circ \eta$ (in $C$), or equivalently $T \eta \circ f = \eta T \circ f$. (The name comes from viewing Kleisli categories as semantics for call-by-value programming languages, see e.g. the references here.)
Every morphism in the image of the free functor $F:C\to C_T$ is thunkable, and the converse holds if $\eta$ is the equalizer of $\eta T$ and $T \eta$. I am told that every nontrivial monad on $\mathrm{Set}$ satisfies this condition.
If $T_1\to T_2$ is a morphism of monads on $C$, there is an induced functor $C_{T_1} \to C_{T_2}$. My question is: is there a morphism of monads such that this induced functor does not preserve thunkability? The above remark shows that if so, $C$ cannot be $\mathrm{Set}$.
