Example 6.2.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms in $\operatorname{\mathcal{C}}$. Applying Corollary 6.2.4.8 in the special case $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_1$, we conclude that $\operatorname{Isom}(\operatorname{\mathcal{C}})$ is a reflective (and coreflective) subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$. Moreover:
A morphism $w$ in $\operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})$ is an $\operatorname{Isom}(\operatorname{\mathcal{C}})$-local equivalence if and only if the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ carries $w$ to an isomorphism in $\operatorname{\mathcal{C}}$.
A morphism $w$ in $\operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}})$ is an $\operatorname{Isom}(\operatorname{\mathcal{C}})$-colocal equivalence if and only if the evaluation functor $\operatorname{ev}_{0}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ carries $w$ to an isomorphism in $\operatorname{\mathcal{C}}$.