nLab 2-category of adjunctions

Idea

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-morphisms are mate-pairs of 2-morphisms 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 1-morphisms of $Adj(K)$ are the objects of $[Adj,K]$ .

The morphisms in $Adj\big(Adj(K)\big)$ are the adjoint triples in $K$ .
The inclusion of Mnd, the free-standing 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 $K$ .

Claude Auderset: Adjonctions et monades au niveau des $2$ -catégories, Cahiers de topologie et géométrie différentielle 15 1 (1974) 3-20 [[numdam:CTGDC_1974__15_1_3_0](https://www.numdam.org/item/CTGDC_1974__15_1_3_0)]
Stephen Schanuel, Ross Street, The free adjunction, Cahiers de topologie et géométrie différentielle catégoriques, 27 1 (1986) 81-83 [[numdam:CTGDC_1986__27_1_81_0](https://www.numdam.org/item/CTGDC_1986__27_1_81_0)]