Let $\mathcal{A}$ be a closed covering of some space $X$, then let $\Sigma[\mathcal{A}]$ be the category of intersections of subsets of $\mathcal{A}$, further let $F: \Sigma[\mathcal{A}] \to \mathsf{Top}$ be the a $\Sigma[\mathcal{A}]$-diagram of spaces with all corresponding maps nice enough (cofibrations and cofibrant images, etc.). Then for some subcategory $J \hookrightarrow \Sigma[\mathcal{A}]$, let
$$ S = \bigcup_{i \in J} \left[F[i] \setminus \bigcup_{j \in \left(i \downarrow \Sigma[\mathcal{A}]\right) } F[j] \right] $$
where $(i \downarrow \Sigma[\mathcal{A}])$ is the undercategory of $\Sigma[\mathcal{A}]$ at $i$. Can we frame this at all as a "relative" colimit or some other categorical construction?
Further we might be able to pull the internal union (the of the setminus) out of the construction by realizing something about the combination of all of these undercategories.
I eventually wish relate $S$ to some sort of relative homotopy colimit, and eventually a combinatorial model in terms of $J$ and $\Sigma[\mathcal{A}]$ so any help would be greatly appreciated.