Remark 1.2.3.14. In the statement of Lemma 1.2.3.13, we can replace $(3)$ by the following pair of conditions:
- $(3')$
For each $n \geq 0$, the standard $n$-simplex $\Delta ^ n$ belongs to $\operatorname{\mathcal{U}}$.
- $(3'')$
The subcategory $\operatorname{\mathcal{U}}\subseteq \operatorname{Set_{\Delta }}$ is closed under the formation of coproducts.