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?



Factorization of colimits through slices?
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.


