Corollary 7.3.3.14 (05E0): Constant Diagrams

Corollary 7.3.3.14 (Constant Diagrams). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which contains an initial object, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:

$(1)$: The functor $F$ is left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{\mathrm{init}} \subseteq \operatorname{\mathcal{C}}$ spanned by the initial objects.
$(2)$: The functor $F$ carries each morphism in $\operatorname{\mathcal{C}}$ to an isomorphism in the $\infty $-category $\operatorname{\mathcal{D}}$.
$(3)$: The functor $F$ is isomorphic to a constant functor.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Corollary 7.3.3.13 by taking $\operatorname{\mathcal{E}}= \Delta ^0$, and the implication $(3) \Rightarrow (2)$ is immediate. To prove the converse, we observe that condition $(2)$ guarantees that $F$ can be regarded as a morphism from $\operatorname{\mathcal{C}}$ to the Kan complex $\operatorname{\mathcal{D}}^{\simeq }$. Since $\operatorname{\mathcal{C}}$ has an initial object, it is weakly contractible (Corollary 4.6.7.25), so this morphism is automatically nullhomotopic (Remark 3.2.4.11). $\square$