Exercise 8.4.1.12. Let $\operatorname{Cat}$ denote the (ordinary) category of small categories, and let $\operatorname{{\bf \Delta }}_{\leq 1} \subset \operatorname{Cat}$ denote the full subcategory spanned by the objects $[0]$ and $[1]$. Show that:
The full subcategory $\operatorname{{\bf \Delta }}_{\leq 1}$ generates $\operatorname{Cat}$ under colimits.
A small category $\operatorname{\mathcal{C}}$ can be realized as the colimit (in $\operatorname{Cat}$) of a diagram $\mathcal{K} \rightarrow \operatorname{{\bf \Delta }}_{\leq 1}$ if and only if the category $\operatorname{\mathcal{C}}$ is free, in the sense of Definition 1.3.7.7.
In particular, the inclusion $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\leq 1} ) \subset \operatorname{N}_{\bullet }( \operatorname{Cat})$ satisfies condition $(3)$ of Warning 8.4.1.10, but does not satisfy condition $(2)$.