Covariant Yoneda embedding
= Covariant Yoneda embedding
{title2=$Y:\mathcal C^{\mathrm{op}}\to[\mathcal C,\mathbf{Set}],\quad Y(A)=\mathcal C(A,-)$}
The covariant version of the <Yoneda embedding> sends an object to its outgoing <representable functor>. Precomposition reverses arrows. The <Yoneda lemma> gives $\operatorname{Nat}(Y(A),Y(B))\cong\mathcal C(B,A)$, so it is <full and faithful>. It carries existing <colimits> in $\mathcal C$ to pointwise <categorical limits>.