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I just came across the following statement while learning Category Theory:

If $\mathcal{C}$ is a small category, then any functor $F: \mathcal{C} \to \mathbf{Set}$ may be expressed as a colimit in $[C, \mathbf{Set}]$ of a diagram of shape $(\mathbf{1}\downarrow F)^{op}$ whose vertices are representable functors, where $\mathbf{1}$ denotes a singleton set.

I can't seem to find a proof for it anywhere, does anyone know how to prove this?

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  • $\begingroup$ This is the "co-Yoneda lemma." See ncatlab.org/nlab/show/co-Yoneda+lemma $\endgroup$ – Qiaochu Yuan Jun 11 '18 at 20:35
  • $\begingroup$ The comma category you refer to is better known as the category of elements of $F$. $\endgroup$ – Derek Elkins Jun 11 '18 at 20:37
  • $\begingroup$ Personally, as somewhat done on the page Qiaochu Yuan mentions, I like to split the result into a result about weighted colimits and another result representing weighted colimits via the category of elements. The weighted colimit/coend approach gives a clean, generalizable result, while the reduction of weighted colimits to conical colimits over the category of elements is a bit of a fluke of $\mathbf{Set}$ and usually gives a more cumbersome formula to work with. $\endgroup$ – Derek Elkins Jun 11 '18 at 20:52

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