Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 6.1.1.7. Let $\operatorname{\mathcal{C}}$ be an ordinary category which admits fiber products, and let $\operatorname{Corr}(\operatorname{\mathcal{C}})$ denote the $2$-category of correspondences in $\operatorname{\mathcal{C}}$ (Example 2.2.2.1). Every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ determines diagrams

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dl]_{\operatorname{id}_ X} \ar [dr]^{f} & & & X \ar [dl]_{f} \ar [dr]^{\operatorname{id}_{X}} & \\ X & & Y & Y & & X } \]

which we can regard as $1$-morphisms $f_{!}: X \rightarrow Y$ and $f^{!}: Y \rightarrow X$ in the $2$-category $\operatorname{Corr}(\operatorname{\mathcal{C}})$. Unwinding the definitions, we see that the compositions $f^{!} \circ f_{!}$ and $f_{!} \circ f^{!}$ are given (up to isomorphism) by the diagrams

\[ \xymatrix@R =50pt@C=50pt{ & X \times _{Y} X \ar [dl]_{\pi _0} \ar [dr]^{\pi _1} & & & X \ar [dl]_{f} \ar [dr]^{f} & \\ X & & X & Y & & Y, } \]

where $\pi _0, \pi _1: X \times _{Y} X \rightarrow X$ are the projection maps. We can therefore regard the diagonal map $\delta : X \rightarrow X \times _{Y} X$ as a $2$-morphism from $\operatorname{id}_ X$ to $f^{!} \circ f_{!}$ in $\operatorname{Corr}(\operatorname{\mathcal{C}})$, and the morphism $f: X \rightarrow Y$ as a $2$-morphism from $f_{!} \circ f^{!}$ to $\operatorname{id}_{Y}$ in $\operatorname{Corr}(\operatorname{\mathcal{C}})$. The pair $( \delta , f )$ is an adjunction between $f_{!}$ and $f^{!}$.