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.