Kerodon

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Corollary 3.3.5.4. Let $f: X \rightarrow Y$ be a Kan fibration of simplicial sets. Then the induced map $\operatorname{Ex}(f): \operatorname{Ex}(X) \rightarrow \operatorname{Ex}(Y)$ is also a Kan fibration.

Proof. We must show that every lifting problem

$\xymatrix@C =50pt@R=50pt{ \Lambda ^{n}_{i} \ar [r] \ar [d] & \operatorname{Ex}(X) \ar [d]^{ \operatorname{Ex}(f) } \\ \Delta ^{n} \ar [r] \ar@ {-->}[ur] & \operatorname{Ex}(Y) }$

admits a solution. This follows by applying Remark 3.1.2.7 to the associated lifting problem

$\xymatrix@C =50pt@R=50pt{ \operatorname{Sd}(\Lambda ^{n}_{i}) \ar [r] \ar [d] & X \ar [d]^{ f} \\ \operatorname{Sd}(\Delta ^{n}) \ar [r] \ar@ {-->}[ur] & Y, }$

since the left vertical map is anodyne by virtue of Proposition 3.3.5.3. $\square$