The Kleisli and Eilenberg–Moore universal properties are schematic

The universal properties of the Kleisli and Eilenberg–Moore adjunctions quantify over each supplied adjunction inducing a fixed monad and assert the existence and uniqueness of particular comparison functors. They do not require a category whose objects are all such adjunctions.

This distinction matters under the library's size convention. The functor category $[\mathcal{C},\mathcal{D}]$ is formed only when $\mathcal{C}$ is small, and for a large source the notation is metatheoretic shorthand rather than a category (Functor category $[\mathcal{C},\mathcal{D}]$); smallness and local smallness of $[\mathcal{C},\mathcal{D}]$ then need the further hypotheses of If $\mathcal{C}$ is small and $\mathcal{D}$ is locally small then $[\mathcal{C},\mathcal{D}]$ is locally small; if both are small it is small. No large category of every monad resolution is presumed here. The following factorisation results state the universal properties in the objectwise form that remains meaningful without such a category.