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Corollary 7.1.5.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. Then:

$(1)$

Let $\overline{u}, \overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a limit diagram if and only if $\overline{v}$ is a limit diagram.

$(2)$

Let $\overline{u}, \overline{v}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a colimit diagram if and only if $\overline{v}$ is a colimit diagram.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $e: \overline{u} \rightarrow \overline{v}$ be an isomorphism in the $\infty $-category $\operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$. Under the canonical isomorphism

\[ \operatorname{Fun}( \Delta ^1, \operatorname{Fun}( K^{\triangleleft },\operatorname{\mathcal{C}}) ) \simeq \operatorname{Fun}( \Delta ^1 \times K^{\triangleleft }, \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( K^{\triangleleft }, \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \]

we can identify $e$ with a diagram $\overline{w}: K^{\triangleleft } \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$, which factors through the full subcategory $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms in $\operatorname{\mathcal{C}}$. Note that the evaluation maps

\[ \operatorname{ev}_0: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}} \]

are trivial Kan fibrations (Corollary 4.4.5.10), and therefore equivalences of $\infty $-categories (Proposition 4.5.2.10). By construction, we have $\overline{u} = \operatorname{ev}_0 \circ \overline{w}$ and $\overline{v} = \operatorname{ev}_1 \circ \overline{w}$. Applying Proposition 7.1.5.5, we deduce that $\overline{u}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $\overline{w}$ is a limit diagram in $\operatorname{Isom}(\operatorname{\mathcal{C}})$. Similarly, $\overline{v}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $\overline{w}$ is a limit diagram in $\operatorname{Isom}(\operatorname{\mathcal{C}})$. $\square$