Example 7.3.9.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram indexed by a weakly contractible simplicial set $K$. Suppose that $f$ carries each edge of $K$ to an isomorphism in $\operatorname{\mathcal{C}}$. Then an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if it carries each edge of $K^{\triangleright }$ to an isomorphism in $\operatorname{\mathcal{C}}$ (this follows by applying Corollary 7.3.9.4 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$). Note that such an extension always exists: the weak contractibility of $K$ guarantees that $f$ is nullhomotopic when viewed as a morphism from $K$ to the core $\operatorname{\mathcal{C}}^{\simeq }$.
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