Corollary 8.1.6.10 (Invertible Cospans). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Let $L$ and $R$ be collections of morphisms of $\operatorname{\mathcal{C}}$ which contain all isomorphisms and are closed under composition. Assume that $L$ and $R$ are pushout-compatible and let $e: X \rightarrow Y$ be a morphism in the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$, which we identify with a diagram $X \xrightarrow {f} B \xleftarrow {g} Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $e$ is an isomorphism in $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ if and only if $f$ and $g$ are isomorphisms in $\operatorname{\mathcal{C}}$.
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Proof. Let $\operatorname{\mathcal{C}}^{\simeq }$ denote the core of $\operatorname{\mathcal{C}}$ (Construction 1.3.5.4). If $f$ and $g$ are isomorphisms, then $e$ can be regarded as an edge of the simplicial set $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\simeq } )$. Since $\operatorname{\mathcal{C}}^{\simeq }$ is a Kan complex (Corollary 4.4.3.11), the simplicial set $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\simeq } )$ is also a Kan complex (Corollary 8.1.3.11), so $e$ is automatically an isomorphism when viewed as a morphism of $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\simeq } )$ (Proposition 1.4.6.10). Since the inclusion map $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\simeq } ) \hookrightarrow \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ is a functor of $(\infty ,2)$-categories (Corollary 8.1.6.8), it follows that $e$ is also an isomorphism when regarded as a morphism of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$ (Remark 5.4.7.5).
Now suppose that $e$ is an isomorphism in the $(\infty ,2)$-category $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$. Arguing as in the proof of Proposition 5.4.6.6, we can produce a $3$-simplex $\sigma : \Delta ^3 \rightarrow \operatorname{Pith}( \operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}}) )$, where $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is the morphism $e$, $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 2 \} )}$ is the identity morphism $\operatorname{id}_{X}$, and $\sigma |_{ \operatorname{N}_{\bullet }( \{ 1 < 3 \} ) }$ is the identity morphism $\operatorname{id}_{Y}$. Let us identify $\sigma $ with a diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [dr]^{f} & & Y \ar [dl]^{\sim }_{g} \ar [dr] & & X \ar [dr] \ar [dl] & & Y \ar [dl] \\ & B \ar [dr]^{u} & & C \ar [dl] \ar [dr] & & D \ar [dl] & \\ & & X \ar [dr]^{v} & & Y \ar [dl] & & \\ & & & Z & & & } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. This diagram exhibits the identity morphism $\operatorname{id}_{X}$ as a composition of $u$ with $f$. Since $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 3 \} ) }$ is a thin $2$-simplex of $\operatorname{Cospan}^{L,R}(\operatorname{\mathcal{C}})$, the outer rectangular region on the left is a pushout square in $\operatorname{\mathcal{C}}$ (Corollary 8.1.6.8). It follows that the composition of $v$ with $u$ is an isomorphism in $\operatorname{\mathcal{C}}$ (since it is a pushout of the identity morphism $\operatorname{id}_{Y}$; see Corollary 7.6.2.27). Applying the two-out-of-six property to the $3$-simplex of $\operatorname{\mathcal{C}}$ given by the left edge of the diagram, we conclude that $f$ is an isomorphism in $\operatorname{\mathcal{C}}$ (see Proposition 5.4.6.5). A similar argument shows that $g$ is an isomorphism in $\operatorname{\mathcal{C}}$. $\square$