Co-Kleisli and co-Eilenberg–Moore adjunctions have the dual extremal universal properties

Regard $G$ as a monad on the opposite category and apply [L3] there.

An adjunction $L⊣R$ inducing $G$ on $\mathcal{C}$ becomes $R^{\mathrm{o}\mathrm{p}}⊣L^{\mathrm{o}\mathrm{p}}$ inducing that monad on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$, and under this translation the Kleisli and Eilenberg–Moore resolutions of the latter are the co-Kleisli and co-Eilenberg–Moore adjunctions of [L1]–[L2]. A morphism of adjunctions is a functor between their intermediate categories, and passing to opposite categories carries such a functor $K$ to $K^{\mathrm{o}\mathrm{p}}$, which runs between the same two adjunctions in the same order. The two comparison directions are therefore preserved rather than reversed, so the initial property stays with co-Kleisli and the terminal one with co-Eilenberg–Moore.

Transporting the two universal properties of [L3] back along this translation gives, for every supplied adjunction inducing $G$, a unique morphism of adjunctions from the co-Kleisli resolution and a unique morphism of adjunctions to the co-Eilenberg–Moore resolution, with the required equalities over the base categories. This assertion quantifies over supplied data and does not require a category of all such adjunctions.