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Everyone knows that you can define (co)ends as certain (co)equalizers, as described at end. A while ago I added the section Connecting the two definitions, which derives the equalizer description from the weighted-limit description by observing that the weight $\hom_C$ for an end is presented as a certain split coequalizer in $[C^{op} \times C, V]$. Now I’m puzzled, though, because this would seem to imply that ends are split equalizers, and dually, split coequalizers being absolute colimits. But it’s easy to cook up an example where this seemingly can’t happen, say with a pair of categories and functors $F,G$ such that $Nat(F,G) = \int_c [F c, G c]$ is empty but $\prod_c [F c, G c]$ isn’t, so there can’t be any morphism from the latter to the former to make the whole thing a split equalizer.
I must have made a mistake somewhere, and knowing me it’ll be a stupidly obvious one, but can anyone point it out?
Are you sure you mean “split” and not reflexive?
OK, so not one but two stupidly obvious mistakes:
I’ll fix end accordingly. Thanks.
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