Decomposing a large colimit as a pushout of smaller colimits I would like to find a reference in the literature for the following result.  I have it on high authority that it isn't in 'Categories for the Working Mathematician' and I can't find it in Borceux's handbook.  It's a result that I'm confident is true (at least when stated correctly) and is probably second nature to category theorists.  I however am writing for group theorists and so want to reference results thoroughly.
I have a functor $F:\mathcal{C} \rightarrow \mathcal{D}$.  The target category $\mathcal{D}$ is cocomplete.  The source category $\mathcal{C}$ is finite and can be decomposed as the pushout of smaller categories $\mathcal{C}_1\leftarrow\mathcal{C}_0\rightarrow\mathcal{C}_2$.
The functors from these into $\mathcal{D}$ are denoted $F_1,F_0$ and $F_2$ respectively.
I need to take the colimit of $F$ and I think that it can be taken to be the pushout of
$\text{colim}F_1\leftarrow\text{colim}F_0\rightarrow\text{colim}F_2$.
Obviously if $\mathcal{C}$ were constructed from a different colimit rather than a pushout one might expect an analogous result.
Giving a proof is an option, but would be out of context with the rest of the paper and probably consigned to an unread appendix.  Or I could just quote it without proof.  Help, or just opinions would be very welcome.
 A: I'd agree that the result is true, and “well-known” to category-theorists!  Unfortunately I don't know a specific reference, but my best guess would be something like Kelly’s “Elements of Enriched CT”, which proves lots of useful things about (co)ends and weighted (co)limits, which specialise to lots of useful things about limits.  It's also hard not to wonder about the legendary treatise of Chevalley on “all possible properties of limits” that got lost in the mail...
If you can't find a reference, though, the proof can certainly be made pretty short — I used 6 well-spaced or 2 cramped lines, and it can probably be compressed further...
[this was meant to be just a comment, but it got a bit too long]
$\newcommand{\C}{\mathbf{C}} \newcommand{\D}{\mathbf{D}} \DeclareMathOperator{\colim}{colim}$
Edit: For clarification, the precise statement I had in mind is that
$$\colim_{I}\ (\colim_{\C_i}\ F_i)\ \cong\ \colim_{\left( \colim_{I} \C_i \right)} [F_i]_{i \in I}$$
where $I$ is a small cat, $\C_i$ is an $I$-indexed diagram of small cats, $F_i : \C_i \to \D$ is a co-cone of functors, $[F\_i]\_{i \in I}$ denotes the induced cotuple functor $\colim_{I} \C\_i \to \D$, and all the relevant colimits exist in $ \D$.
