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Proposition 5.3.5.10 (Inner Exchange). Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. Then the collection of thin $2$-simplices of $\operatorname{\mathcal{C}}$ has the inner exchange property (Definition 5.3.5.7).

Proof of Proposition 5.3.5.10. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $\sigma : \Delta ^{3} \rightarrow \operatorname{\mathcal{C}}$ be a $3$-simplex of $\operatorname{\mathcal{C}}$ and let $C = \sigma (3) \in \operatorname{\mathcal{C}}$ be the image of the final vertex. Let us regard the face $\sigma _{210} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 2 \} )}$ as a morphism of simplicial sets from $\Delta ^2$ to $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ denote the pullback $\Delta ^{2} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Note that the projection map $\operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration (Proposition 5.3.3.1). If $\sigma _{210}$ is thin, then the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ is also an interior fibration (Remark 5.3.2.4); since $\Delta ^2$ is an $\infty $-category, it is an inner fibration (Example 5.3.2.2). Unwinding the definitions, we can identify $\sigma $ with a $2$-simplex of $\operatorname{\mathcal{E}}$ lying over the unique nondegenerate $2$-simplex of $\Delta ^2$, which we display as a diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]_{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z. } \]

If $\sigma _{321} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 1 < 2 < 3 \} )}$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$, then the “easy direction” of Theorem 5.3.4.1 guarantees that $g$ is $\pi $-cartesian. It follows that $f$ is $\pi $-cartesian if and only if $h$ is $\pi $-cartesian (Corollary 5.1.2.5). Equivalently, $f$ is locally $\pi $-cartesian if and only if $h$ is locally $\pi $-cartesian (see Remark 5.1.3.5). Applying the “hard direction” of Theorem 5.3.4.1, we conclude that the $2$-simplex $\sigma _{310} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 3\} ) }$ is thin if and only if the $2$-simplex $\sigma _{320} = \sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 2 < 3 \} )}$ is thin. $\square$