Covariant Yoneda embedding ($Y:{\mathcal{C}}^{\mathrm{op}}\to [\mathcal{C},\mathbf{Set}],Y(A)=\mathcal{C}(A,-)$)

The covariant version of the Yoneda embedding sends an object to its outgoing representable functor. Precomposition reverses arrows. The Yoneda lemma gives $\mathrm{Nat}(Y(A),Y(B))\cong \mathcal{C}(B,A)$, so it is full and faithful. It carries existing colimits in $\mathcal{C}$ to pointwise categorical limits.