The Clowder Project

The contravariant Yoneda embedding of $\mathcal{C}$ is the functor¹

where

• •

  Action on Objects. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, we have

  \[ \style {display: inline-block; transform: rotate(180deg)}{よ}\mkern -2.5mu_{\mathcal{C}}(A) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h^{A}. \]
• •

  Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the action on morphisms

  \[ \style {display: inline-block; transform: rotate(180deg)}{よ}\mkern -2.5mu_{\mathcal{C}|A,B}\colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)\to \operatorname {\mathrm{Nat}}(h^{B},h^{A}) \]

  of $\style {display: inline-block; transform: rotate(180deg)}{よ}\mkern -2.5mu_{\mathcal{C}}$ at $(A,B)$ is given by

  \[ \style {display: inline-block; transform: rotate(180deg)}{よ}\mkern -2.5mu_{\mathcal{C}|A,B}(f)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}h^{f} \]

  for each $f\in \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)$, where $h^{f}$ is the corepresentable natural transformation associated to $f$ of Definition 12.2.3.1.1.