# A morphism of monads that doesn't preserve thunkability?

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 $$\mathrm{Set}$$.

• Is it trivial to show that every nontrivial monad on Set satisfies the condition? – Ivan Di Liberti May 22 at 20:56
• @IvanDiLiberti I don't know, I haven't tried. – Mike Shulman May 22 at 22:48
• (BTW there are two "trivial" monads on Set: one is constant at 1, the other is constant at 1 everywhere except it sends 0 to 0.) – Mike Shulman May 22 at 22:49