= Canonical colimit presentation of a covariant set-valued functor
{title2=$F\cong\operatorname{colim}_{(1\downarrow F)^{\mathrm{op}}}\mathcal C(A,-)$}
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.
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