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I could swear I remember a result of the following form:

Suppose we have a pair of functors $$C\xrightarrow{F}D\xrightarrow{G} X,$$ with $X$ cocomplete.

then we obtain a functor $$D\to X$$ sending $$d\mapsto \operatorname{colim}_{(F\downarrow d)}(G\circ \pi_d)$$ where $\pi_d:(F\downarrow d)\to D$ is the projection functor sending $(F(c)\to d) \mapsto F(c)$.

Then there is a canonical isomorphism $$\operatorname{colim}_D\left (\operatorname{colim}_{(F\downarrow d)}(G\circ \pi_d)\right) \cong \operatorname{colim}_C (G\circ F).$$

Is this true? Do you know of a source? Is there a name for this kind of result?

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  • $\begingroup$ The closest thing I can find to this is HTT 4.2.3, which considers a more general situation but can ostensibly be used in this situation. I'm looking for a more classical reference, since this seems like it should be an older result. $\endgroup$ Sep 17 '18 at 1:20
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    $\begingroup$ Probably it should be noted, that $\text{colim}_{(F\downarrow d)}(G\circ\pi_d)$ is a pointwise left Kan extension of $G\circ F$ along $F$. $\endgroup$
    – Oskar
    Sep 17 '18 at 1:31
  • $\begingroup$ @Oskar Doesn't that prove the statement then, since $\operatorname{colim}(\operatorname{Lan}_F G\circ F)=\operatorname{Lan}_{t_D}\left(\operatorname{Lan}_F G\circ F \right)=\operatorname{Lan}_{t_D \circ F} G\circ F =\operatorname{Lan}_{t_C}G\circ F=\operatorname{colim} G\circ F$ since the terminal functor $t_C=t_D\circ F$? $\endgroup$ Sep 17 '18 at 2:00
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    $\begingroup$ @HarryGingi Yes, seems that it proves this result. $\endgroup$
    – Oskar
    Sep 17 '18 at 2:23
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    $\begingroup$ I added an answer, it is a summary of the comments. $\endgroup$
    – Oskar
    Sep 17 '18 at 4:07
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The functor $\text{colim}_{(F\downarrow -)}(G\circ\pi_-)\colon D\to X$ is a left Kan extension of $G\circ F$ along $F$. The corresponding natural transformation $\ell_F^{G\circ F}\colon G\circ F\to \text{colim}_{(F\downarrow -)}(G\circ\pi_-)\circ F$ is defined by $$ \ell_F^{G\circ F}(c)=\varphi^{F(c)}(id_{F(c)}), $$ where $\varphi^{F(c)}$ is a colimiting cocone of $G\circ\pi_{F(c)}$. Verification of the universality is long but straightforward.

Then it remains to note that for any functors $T$ and $S$ there exists an isomorphism $$ \varinjlim\text{Lan}_TS\cong\varinjlim S. $$

Seems that your statement is just one of the great amount of interesting properties of Kan extensions, so I doubt that it has a name. Some basic properties of pointwise Kan extensions are described in the first volume of Borceux's handbook.

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    $\begingroup$ This is in fact a result about compatibility of different weights in weighted colimits, valid more generally in enriched/formal category theory. See for example Riehl's Categorical Homotopy Theory, Lemma 8.1.4. $\endgroup$
    – fosco
    Sep 17 '18 at 20:40

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