Let $g := \underset{c\in ob(\mathcal{C})}{\coprod} \frac{\mathcal{F}(c)}{\sim}$ for a small category $\mathcal{C}$ and $g \in Top$ where $\sim$ is the equivalence relation $x\sim\mathcal{F}(f)x$ for $x\in\mathcal{F}(c_{1})\ni$
$$f:c_{1}\rightarrow c_{2}$$ so $$\mathcal{F}(f)x\in\mathcal{F}(c_{2})$$
with functor $\mathcal{F}:\mathcal{C}\rightarrow Top$.
This constructs the canonical map $$\varphi_{\mathcal{F}}(c):\mathcal{F\Longrightarrow g}$$
with natural transformation:
$$\varphi_{\mathcal{F}}:\mathcal{F\Longrightarrow \mathcal{F}_g}$$
So: $$g=\underset{\mathcal{C}}{colim}\mathcal{F}$$
I understand that a coproduct is the colimit over the diagram that consists of just two objects. So is the reason that $g$ is the colimit just precisely by construction where the diagram here would successively have $f$ edges between nodes of $c$?
Is there something additional that would need to be demonstrated to substantiate the final statement (that $g$ is $\underset{\mathcal{C}}{colim \mathcal{F}}$)?