Double Category of Mates

The double category of makes in a 2-Category $K$ is the Double Category $\mathrm{Adj}(K)$ with

• Objects are objects of $K$

• Tight morphisms $A\to B$ are 1-cells of $K$

• Loose morphisms $A\slashedrightarrow B$ are Adjoints $L:A\to B,R:B\to A,\eta:LR\Rightarrow mathrm{Id}{ifx..lse\par i},\eta:mathrm{ Id}{ifx..lse\par i}\Rightarrow RL$ in $K$

• 2-Cells are Mate Pairs: \[

  \begin{tikzcd}

A && B

C && D \end{tikzcd} \]