For any 2-category$K$, there are two 2-categories (each with several variants) that could be called “the 2-category of adjunctions in $K$”.

The 2-category which here we call $Adj(K)$ has the same objects as $K$, its morphisms are the adjunctions in $K$ (pointing in the direction of, say, the left adjoint), and its 2-cells are mate-pairs of 2-cells between adjunctions in $K$.

The 2-category $[Adj,K]$ is the functor 2-category from the walking adjunction$Adj$ to $K$. Thus its objects are the adjunctions in $K$ — or more precisely, triples $(x,y,(f,g,\eta,\epsilon))$ where $x,y$ are objects of $K$ and $(f,g,\eta,\epsilon)$ is an adjunction between $x$ and $y$. Its morphisms are pairs of morphisms $x\to x'$ and $y\to y'$ such that certain squares commute (perhaps up to a transformation or isomorphism), and its 2-cells are similarly composed of cylinders.

Note that the morphisms of $Adj(K)$ are the objects of $[Adj,K]$.

Properties

The morphisms in $Adj(Adj(K))$ are the adjoint triples in $K$.

The inclusion of $Mnd$, the free monad, in $Adj$ induces a 2-functor from $[Adj,K]$ to $[Mnd,K]$, the 2-category of monads in $K$. The adjoints to this 2-functor are the Kleisli and Eilenberg-Moore constructions on monads in $K$.

References

Stephen Schanuel and Ross Street, The free adjunction, Cahiers de topologie et géométrie différentielle catégoriques, tome 27, no 1 (1986), p. 81-83

Last revised on July 7, 2021 at 18:46:44.
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