Let \(h^X = \mathrm{Hom}(X,)\) and \(h_Y=\mathrm{Hom}(,Y)\) be the covariant and contravariant \(\mathrm{Hom}\) functors. The Yoneda’s Lemma states, in the category of (covariant and contravariant) functors, for a covariant functor \(F:\mathcal{C}\to {\bf Sets}\) and contravariant functor \(G:\mathcal{C}^{op}\to {\bf Sets}\), we have
\(\mathrm{Hom}(h^X, F) = F(X)\)
\(\mathrm{Hom}(h_Y, G) = G(Y).\)
As a special case we have
\(\mathrm{Hom}(h_X,h_Y) = \mathrm{Hom}(X,Y)\)
\(\mathrm{Hom}(h^X, h^Y) = \mathrm{Hom}(Y,X)\)
i.e. \(h_\square : \mathcal{C}\to \mathrm{Hom}(\mathcal{C}^{op},{\bf Sets})\) and \(h^\square: \mathcal{C}^{op}\to \mathrm{Hom}(\mathcal{C}, {\bf Sets})\) are fully faithful functors.