Example 7.7.1.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K = \emptyset $ be the empty simplicial set. Then we can identify a diagram $\overline{\mathscr {F}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ which an object $C \in \operatorname{\mathcal{C}}$. In this case, the following conditions are equivalent:
- $(a)$
The morphism $\overline{\mathscr {F}}$ is a universal colimit diagram.
- $(b)$
For every morphism $u: C' \rightarrow C$ of $\operatorname{\mathcal{C}}$, the object $C' \in \operatorname{\mathcal{C}}$ is initial.
- $(c)$
The object $C \in \operatorname{\mathcal{C}}$ is initial and every morphism $u: C' \rightarrow C$ is an isomorphism.
If these conditions are satisfied, we say that $C$ is a universal initial object of $\operatorname{\mathcal{C}}$.