Adjunction in a \( 2 \)-Category
For any \( 2 \)-category \( \mathcal C \) we can define what an adjunction is in \( \mathcal C \) using the Adjunctions via Unit and Counit, because this definition canonically lives in the \( 2 \)-world. Then an adjunction in the strict \( 2 \)-category \( \Cat \) is just a normal adjunction between functors.
Let \( \mathcal C \) be a \( 2 \)-category. An adjunction in \( \mathcal C \) consists of
- two \( 1 \)-morphisms \( l \colon x \rightarrow y \) and \( r \colon y \rightarrow x \),
- a \( 2 \)-morphism \( \eta \colon 1_x \rightarrow rl \) (unit),
- a \( 2 \)-morphism \( \epsilon \colon lr \rightarrow 1_y \) (counit),
such that
\begin{equation*} \xymatrix{ x \ar@{->}[r]_{l} \ar@/^1.5pc/@{->}[rr]^{1_x}="1" & y \ar@{->}[r]^{r} \ar@/_1.5pc/@{->}[rr]_{1_y}="2" & x \ar@{->}[r]^{l} & y \ar@{=>}"1";"1,2"^{\eta} \ar@{=>}"1,3";"2"^{\epsilon}} = \xymatrix{ x \ar@/^1.5pc/[r]^l="1" \ar@/_1.5pc/[r]_l="2" & y \ar@{=>}"1";"2"^{1_l}} \end{equation*}and
\begin{equation*} \xymatrix{ y \ar@{->}[r]^{l} \ar@/_1.5pc/@{->}[rr]_{1_y}="2" & x \ar@{->}[r]^{r} \ar@/^1.5pc/@{->}[rr]^{1_x}="1" & y \ar@{->}[r]_{l} & x \ar@{=>}"1";"1,3"^{\eta} \ar@{=>}"1,2";"2"^{\epsilon}} = \xymatrix{ y \ar@/^1.5pc/[r]^r="1" \ar@/_1.5pc/[r]_r="2" & x \ar@{=>}"1";"2"^{1_r}} \end{equation*}