Corollary 8.1.3.10. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Kan fibration of simplicial sets. Then the induced map $\operatorname{Cospan}(U): \operatorname{Cospan}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$ is also a Kan fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets which is a weak homotopy equivalence. We wish to show that every lifting problem
8.5
\begin{equation} \begin{gathered}\label{equation:cospan-Kan-fibration} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{i} & \operatorname{Cospan}(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{Cospan}(U) } \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{Cospan}(\operatorname{\mathcal{D}}) } \end{gathered} \end{equation}
admits a solution. Using Proposition 8.1.3.7, we can rewrite (8.5) as a lifting problem of the form
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(A) \ar [r] \ar [d]^{\operatorname{Tw}(i)} & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{Tw}(B) \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}. } \]
Our assumption that $U$ is a Kan fibration guarantees that this lifting problem has a solution, since the monomorphism $\operatorname{Tw}(i): \operatorname{Tw}(A) \hookrightarrow \operatorname{Tw}(B)$ is also a weak homotopy equivalence (Corollary 8.1.2.6). $\square$