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.
and it's also denoted by \(\left [{\cal C}^{op}, \mathbf {Set}\right ]\left (\mathscr {H}_X, \mathscr {F}\right )\) in the sense of \(\operatorname {Hom}_{\left [{\cal C}^{op}, \mathbf {Set}\right ]}\left (\mathscr {H}_X, \mathscr {F}\right )\) where \(\left [{\cal C}^{op}, \mathbf {Set}\right ]\) is a functor category.