Kerodon

Proof. Let $\sigma $ be a thin $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$; we will show that $\sigma $ is also thin when viewed as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ (the reverse implication follows from Remark 8.1.6.4). Choose a fully faithful functor $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is an $\infty $-category which admits pushouts and the functor $f$ preserves all pushout squares which exist in $\operatorname{\mathcal{C}}$ (see Corollary 8.3.3.17). Then $f$ induces a morphism of simplicial sets $F: \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$. The proof of Proposition 8.1.6.7 shows that every morphism $\tau _0: \Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ can be extended to a $2$-simplex $\tau $ of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ which is thin when viewed as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. Since $f$ preserves pushout squares, the criterion of Proposition 8.1.4.2 guarantees that $F(\tau )$ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{D}})$. Allowing $\tau _0$ to vary and invoking Proposition 5.4.7.9, we deduce that $F$ is a functor of $(\infty ,2)$-categories. In particular, $F(\sigma )$ is a thin $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{D}})$. Using the criterion of Proposition 8.1.4.2 and the assumption that $f$ is fully faithful, we deduce that $\sigma $ is also thin when viewed as a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. $\square$