# Kerodon

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Lemma 8.1.7.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{D}}$ be a simplicial set, and suppose we are given a pair of diagrams $f,g: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}})$, corresponding to diagrams $F,G: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The diagrams $f$ and $g$ are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) )$.

$(2)$

The diagrams $F$ and $G$ are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$.

Proof. Let $\operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ spanned by those functors $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ which correspond to diagrams $\operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso} }(\operatorname{\mathcal{C}})$. Lemma 8.1.7.12, identifies $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{\mathcal{C}}) )$ with the $\infty$-category $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$. We are therefore reduced to proving that $F$ and $G$ are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}})$ if and only if they are isomorphic when viewed as objects of the $\infty$-category $\operatorname{Cospan}^{\mathrm{all}, \mathrm{iso}}( \operatorname{Fun}'( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) )$. This is a special case of Corollary 8.1.6.12. $\square$