$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark We can state Corollary more informally as follows: the category $\operatorname{Set_{\Delta }}$ of simplicial sets is generated, under small colimits, by objects of the form $\Delta ^ n$. In fact, one can say more: it is freely generated (under small colimits) by the essential image of the Yoneda embedding

\[ \operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Set_{\Delta }}\quad \quad [n] \mapsto \Delta ^ n. \]

This is a general fact about presheaf categories: see Proposition (and Theorem for an analogous statement in the setting of $\infty $-categories).