Canonical colimit presentation of a covariant set-valued functor ($F\cong {\mathrm{colim}}_{(1\downarrow F{)}^{\mathrm{op}}}\mathcal{C}(A,-)$)

Every functor $F:\mathcal{C}\to \mathbf{Set}$ on a small category is the colimit of the covariant representable functors $\mathcal{C}(A,-)$ indexed by the opposite of its category of elements. The cocone indexed by $(A,x)$ sends $f:A\to B$ to $F(f)(x)$. By the Yoneda lemma, a compatible cocone into $G$ is exactly a natural transformation $F\to G$. The opposite indexing direction is essential because precomposition reverses the representing-object arrow.