From monads to comonads by the calculus of mates If a comonad $D$ is left adjoint to an endofunctor $T$, then $T$ can be made into a monad: its unit and multiplication are given respectively by the mates of the counit and comultiplication of $D$.
Proposition the co-Kleisli category of $D$ is isomorphic to the Kleisli category of $T$.
Proof. This comes from the adjunction isomorphism $\mathbb C(D \_, \_) = \mathbb C(\_, T \_)$ $\quad \Box$.
Similarly (or dually, in one of the three 2-categorical senses op, co, or coop), if a monad $T'$ is left adjoint to an endofunctor $D'$, then $D'$ can be made into a comonad: It is easy to check that its counit and comultiplication are given respectively by the mates of the unit and multiplication of $T'$.
Question: In this case, what is the corresponding proposition to the above one? How is it proved?
My own attempt: I have tried to show that the Kleisli category of $T'$ is isomorphic to the co-Kleisli category of $D'$. I could easily prove it if I had an isomorphism $\mathbb C(\_, T' \_) = \mathbb C(D' \_, \_)$. But the adjunction isomorphism is $\mathbb C(T' \_, \_) = \mathbb C(\_, D' \_)$.
 A: You have not dualised the statement correctly.
According to your comments (now deleted) you wish to consider Kleisli objects in $\textbf{Cat}^\textrm{op}$ (and $\textbf{Cat}^\textrm{coop}$).
If you write out the definition in full, you will discover that they are actually Eilenberg–MacLane categories.
Thus the correct dualisation of the statement is:

If a monad $T'$ is left adjoint to an endofunctor $D'$, then $D'$ can be made into a comonad and the Eilenberg–MacLane category of $T'$-algebras is isomorphic to the Eilenberg–MacLane category of $D'$-coalgebras.

As it turns out, this is a true statement, but whether your proof dualises or not depends on the details of the proof.
In fact, I would say that this version is more fundamental, in the same way that statements about limits in $\textbf{Set}$ are more fundamental than statements about colimits in arbitrary categories.

Let us recall the definition of Kleisli object.
Given a monad $\mathbb{T} = (T, \eta, \mu)$ on an object $C$ in a 2-category, a Kleisli object of $\mathbb{T}$ is an object $C_\mathbb{T}$ equipped with a morphism $f_\mathbb{T} : C \to C_\mathbb{T}$ and a 2-cell $\alpha_\mathbb{T} : f_\mathbb{T} t \Rightarrow f_\mathbb{T}$ such that:

*

*$\alpha_\mathbb{T} \bullet f_\mathbb{T} \eta = \textrm{id}_{f_\mathbb{T}}$ and $\alpha_\mathbb{T} \bullet \alpha_\mathbb{T} t = \alpha_\mathbb{T} \bullet f_\mathbb{T} \mu$.


*$(f_\mathbb{T}, \alpha_\mathbb{T})$ is initial among all such pairs, i.e. given a morphism $f : C \to D$ and a 2-cell $\alpha : f t \Rightarrow f$ such that $\alpha \bullet f \eta = \textrm{id}_f$ and $\alpha \bullet \alpha t = \alpha \bullet f \mu$, there is a unique morphism $g : C_\mathbb{T} \to D$ such that $f = g \circ f_\mathbb{T}$ and $\alpha = g \alpha_\mathbb{T}$.
(There is also a condition on 2-cells between such pairs, but I omit the details.)
You can verify that the usual Kleisli category of a monad in $\textbf{Cat}$ has the above universal property.
Now, let us dualise to $\textbf{Cat}^\textrm{op}$.
Monads in $\textbf{Cat}^\textrm{op}$ are monads in $\textbf{Cat}$.
A Kleisli object in $\textbf{Cat}^\textrm{op}$ is then a category $\mathcal{C}^\mathbb{T}$ equipped with a functor $u^\mathbb{T} : \mathcal{C}^\mathbb{T} \to \mathcal{C}$ and a natural transformation $\alpha^\mathbb{T} : t u_\mathbb{T} \Rightarrow u_\mathbb{T}$ such that:

*

*$\alpha^\mathbb{T} \bullet \eta u^\mathbb{T} = \textrm{id}_{u^\mathbb{T}}$ and $\alpha^\mathbb{T} \bullet t \alpha^\mathbb{T} = \alpha^\mathbb{T} \bullet \mu u^\mathbb{T}$.


*$(u^\mathbb{T}, \alpha^\mathbb{T})$ is terminal among all such pairs.
You can verify that the usual Eilenberg–MacLane category of algebras of a monad in $\textbf{Cat}$ has this universal property.
