Yoneda embedding

{{category theory}} Given category <math>\mathcal{C}</math>, a '''Yoneda embedding''' for this category is a functor <math>\phi</math> such that for any object ''A'' in <math>\mathcal{C}</math>, <math> \phi: A \mapsto h^A </math> and for any morphism <math> f:B \rightarrow A </math> in <math>\mathcal{C}</math>, <math> \phi: f \mapsto \eta: h^A \rightarrow h^B </math> where the natural transformation ''&eta;'' has components <math> \eta_X: s \mapsto s\circ f </math>. Then <math> \phi: \mathcal{C}^{op} \rightarrow [\mathcal{C},\mathcal{S}ets]</math>. Otherwise, it is a functor <math>\phi</math> such that <math> \phi: A \mapsto h_A </math> and for any <math> f:A \rightarrow B </math> in <math>\mathcal{C}</math>, <math> \phi: f \mapsto \eta: h_A \rightarrow h_B </math> where ''&eta;'' has components <math> \eta_X: s \mapsto f\circ s </math>. Then <math> \phi: \mathcal{C} \rightarrow [\mathcal{C}^{op}, \mathcal{S}ets] </math>.