Remark 8.1.7.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $R$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which has the left two-out-of-three property, and let $\operatorname{\mathcal{D}}$ be a simplicial set. Suppose we are given a pair of morphisms $f,g: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}^{ \mathrm{all}, R}( \operatorname{\mathcal{C}})$, corresponding to diagrams $F,G: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$. If $\alpha : F \rightarrow G$ is a natural transformation of diagrams, then the following conditions are equivalent:
- $(a)$
For every edge $u: D' \rightarrow D$ of $\operatorname{\mathcal{D}}$, the morphism $\alpha _{u}: F(u) \rightarrow G(u)$ belongs to $R$.
- $(b)$
For every degenerate edge $u: D \rightarrow D$ of $\operatorname{\mathcal{D}}$, the morphism $\alpha _{u}: F(u) \rightarrow G(u)$ belongs to $R$.
The implication $(a) \Rightarrow (b)$ is immediate. To prove the reverse implication, let $u: D' \rightarrow D$ be an edge of $\operatorname{\mathcal{D}}$ and let $u_{R}: \operatorname{id}_{D} \rightarrow u$ be the edge of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ described in Example 8.1.3.6. Evaluating $\alpha $ on the morphism $u_{R}$, we obtain a commutative diagram
in the $\infty $-category $\operatorname{\mathcal{C}}$, where the vertical maps belong to $R$ by virtue of our assumption that $f$ and $g$ factor through $\operatorname{Cospan}^{\mathrm{all}, R}(\operatorname{\mathcal{C}})$. Applying the left two-out-of-three property, we conclude that if the upper horizontal map belongs to $R$, then the lower horizontal map also belongs to $R$.