$X$ a $\Delta$-complex with simplices $(\sigma_\alpha\colon \Delta^{n_\alpha} \to X)_{\alpha}$, then
\begin{equation*}
\left(\coprod_\alpha \Delta^{n_\alpha}\right)\big/{\sim} \cong X
\end{equation*}
where $\Delta^{n_\alpha} \ni x \sim y \in \Delta^{n_\beta}$ iff $\sigma_\alpha(x)=\sigma_\beta(y)$.
The bijection is given by $\Delta^{n_\alpha} \ni [x] \mapsto \sigma_\alpha(x)$ and (iii) tells you this is a homeomorphism.