definition. Yoneda embedding functors [leinster2016basic, 4.1.15] [tt-002T]

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Let \({\cal C}\) be a locally small category. The covariant Yoneda embedding functor of \({\cal C}\) is the functor \[ \mathscr {H}^{\bullet }: {\cal C}^{op} \to [{\cal C}, \mathbf {Set}] \] defined on objects \(X\) by the covariant hom-functor on \(X\).

This functor embeds what every object in \({\cal C}\) sees the "world" of the category \({\cal C}\), i.e. arrows from each object.

Conversely, the (contravariant) Yoneda embedding functor of \({\cal C}\) is the functor \[ \mathscr {H}_{\bullet }: {\cal C} \to [{\cal C}^{op}, \mathbf {Set}] \] defined on objects \(X\) by the contravariant hom-functor on \(X\).

This functor embeds how every object in \({\cal C}\) is "seen", i.e. arrows to each object.

\(\bullet \) is a placeholder for an object. \(\mathscr {H}^X\) and \(\mathscr {H}_X\) denote the corresponding Yoneda embedding functors applied to \(X\), and are called covariant/contravariant Yoneda functors, respectively.

Diagramatically [rosiak2022sheaf, def. 161]:

When one speaks of the Yoneda (embedding) functor without specifying covariant or contravariant, it means the contravariant one, because it's the one used in the Yoneda lemma.