In the category of topological spaces, I would like to know that quotients of simplicial complexes (or $\Delta$-complexes) by equivalence relations which are "unramified" in a suitable sense still have the structure of simplicial complexes:

Say $X$ and $Y$ are finite disjoint unions of simplices (of varying dimension) and we have two maps $X \rightrightarrows Y$ such that restricted to each simplex $\Delta^k_i \subset X$, each map is a linear closed embedding into one of the simplices $\Delta^n_j \subset Y$. By linear closed embedding I mean that the map $\Delta^k_i \to \Delta^n_j$ is the inclusion of the convex hull of $k$ linearly independent points (which for my applications can be assumed to have rational coordinates) in $\Delta^n_j$.

It seems plausible that such a diagram $X \rightrightarrows Y$ can be identified with the geometric realization of a diagram of finite $\Delta$-sets, which would equip the coequalizer $coeq(X \rightrightarrows Y)$ with the structure of a finite $\Delta$-complex. Is there a counterexample to this claim that I'm missing? Does this appear in the literature?