Isomorphisms of Kleisli categories

Question

Let $\mathsf{T}_1=(T_1,\eta_1,\mu_1)$ and $\mathsf{T}_2=(T_2,\delta_2,\mu_2)$ be monads on a category $\mathcal{C}$. We say that an isomorphism $\delta\colon \mathsf{T}_1\to \mathsf{T}_2$ of monads is a natural isomorphism $\delta\colon T_1\to T_2$ satisfying $\mu_2\circ \delta^2 = \delta \circ \mu_1$ and $\eta_2 = \delta\circ \eta_1$, where $\delta^2 = T_2(\delta)\circ \delta_{T_1}= \delta_{T_2}\circ T_1(\delta)$.

Obviously, given such an isomorphism $\delta$, it induces an isomorphism $E_{\delta}\colon \mathcal{C}_{\mathsf{T}_1} \to \mathcal{C}_{\mathsf{T}_2}$ of the Kleisli categories. Explicitely, $E_{\delta}$ acts as identity on objects, and sends a morphism $f\colon C_1\to T_1(C_2)$ to $E_{\delta}(f)=\delta \circ f\colon C_1\to T_2(C_2)$. If we let $F_{\mathsf{T}_1}\colon \mathcal{C}\to \mathcal{C}_{\mathsf{T}_1}$ and $F_{\mathsf{T}_2}\colon \mathcal{C}\to \mathcal{C}_{\mathsf{T}_2}$ denote the left adjoints in the Kleisli adjunction, then we also have that $E_{\delta}\circ F_{\mathsf{T}_1}=F_{\mathsf{T}_2}$.

I believe the following statement should also be true; Given an isomorphism $E\colon \mathcal{C}_{\mathsf{T}_1} \to \mathcal{C}_{\mathsf{T}_2}$ of the Kleisli categories satisfying $E\circ F_{\mathsf{T}_1}=F_{\mathsf{T}_2}$, then there exist a unique natural isomorphism $\delta\colon \mathsf{T}_1\to \mathsf{T}_2$ of monads such that $E_{\delta}=E$.

Does anyone know a reference for this result? There probably exist a more general result which implies what I stated above.

Thanks for the help!

Answer 1

The more general result is that if $C$ is an object of a 2-category $K$ that admits Eilenberg-Moore objects, then the induced functor $\mathrm{EM} : \mathrm{Monads}(C) \to (K/C)^{\mathrm{op}}$ is fully faithful. This follows from Theorem 6 of Ross Street's The formal theory of monads, which says that this functor in fact has a partial right adjoint with invertible unit, so it is the inclusion of a coreflective subcategory of a full subcategory. (Street says there is a partial left adjoint to $\mathrm{Monads}(C) \to K/C$, but his category of monads has reversed arrows from the obvious ones, since it is the category of lax monad morphism structures on the identity functor.)

In particular, an isomorphism between Eilenberg-Moore objects must come from an isomorphism of monads. Applying this in $\mathrm{Cat}^{\mathrm{op}}$, whose Eilenberg-Moore objects are Kleisli categories, and whose slice categories are co-slice categories of $\mathrm{Cat}$, yields your fact.

Of course, none of this high technology is necessary to prove your fact; one can simply observe that $E$ must also commute with the right adjoints $U$ of the Kleisli adjunction and that $T = U_T F_T$. But you asked for a reference and a more general result.