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!