Monad, comonad and adjunction are 2-categorical notions. Results about them can be dualized as shown in this answer.
In the second part the answer, the dualization is successfully applied to the special case of small categories. It is possible because they indeed form a 2-category. But how to deal with locally small categories? The problem I see with them is that they do not form a 2-category.
About the terminology used in this question:
A category is small iff each of its hom-sets is a set and it has a set of objects. It is locally small iff each of its hom-sets is a set but its collection of objects is a proper class.
In the context of category theory in ZF(C), a universe is a set that's closed under all the operations of ZF(C). That is, it contains an empty set and is closed under taking unions, intersections, powersets, etc. Effectively, it behaves like a "set of all sets", even though it doesn't literally contain every set (it doesn't contain itself, for one).
One way to introduce universes in this context is to add the axiom "every set is contained in some universe". This ensures that we have enough universes to cover any arbitrarily large (but still set-sized) constructions.
In this framework, all categories are small in the traditional sense. We don't have a category of all sets, of all groups, etc. Rather than talking about proper classes as is traditional, we simply fix some universe talk about things relative to that universe. So for example, rather than a category of all sets, we have a category of $V$-small sets, meaning the category of sets contained in the universe $V$.
Of course, the set of $V$-small sets isn't itself $V$-small, but it is $V'$-small for some slightly larger universe $V'$. So whenever we would normally talk about proper classes, we can simply raise the universe one step instead.
Since every set is contained in some universe, if we want to include some arbitrary set in this category, we just need to make $V$ large enough. Keep in mind that any universe is already incomprehensibly large and it's fair to say that any set used in "ordinary" mathematics already appears in the smallest universe (or the smallest that contains the natural numbers, depending on your exact definition of universe).
So rather than talk about the 2-category of all small categories, we fix a universe $V$ and instead talk about the 2-category of $V$-small categories. This means that the set of objects and the set of morphisms must both be contained in $V$. For the usual reasons, this forms a strict 2-category. The category of $V$-small sets isn't $V$-small, but it is $V'$-small for some slighly larger universe $V'$.
Similarly, rather than talking about locally small categories, we fix a universe $V$ and talk about locally $V$-small categories. These are categories whose hom sets are contained in $V$. Of course, in addition to constraining the size of the hom sets, we might also want to constraint the size of the category as a whole. Fixing another universe $U$, we can talk about $U$-small categories that are also locally $V$-small. Following the usual proof, this also forms a strict 2-category. It isn't a $U$-small category, but it is $U'$-small for some slightly larger $U'$.
Note that $V$-small categories are automatically locally $V$-small, but we probably want more control than that. For example, the category of $V$-small sets is locally $V$-small, even though the category itself isn't $V$-small. That's why we have two universe parameters in $V$-small, locally $U$-small categories.
So say we have some theorem about the objects of 2-categories that usefully dualizes and we want to apply this theorem to some particular locally small category $\mathcal C$.
First, we realize that in our framework, this category is $V$-small for some universe $V$. Even if it's supposed to be the category of sets, we really just talk about the category of $U$-small sets.
Second, we can refine the notion of locally small, to locally $U$-small. Any particular category has all its hom sets in some universe, so let's give a name to that universe.
Finally, we apply our theorem to the 2-category of $V$-small, locally $U$-small categories, as well as its $^{op}$, its $^{co}$ and its $^{coop}$, giving us four statements about $\mathcal C$ for the price of one.