$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 4.3.7.15. Let $f: K \rightarrow X$ and $q: X \rightarrow S$ be morphisms of simplicial sets, let $K_0 \subseteq K$ be a simplicial subset, and set $f_0 = f|_{K_0}$. If $q$ is a trivial Kan fibration, then the induced maps

\[ X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)} \quad \quad X_{f/} \rightarrow X_{f_0/} \times _{ S_{(q \circ f_0)/ } } S_{(q \circ f)/} \]

are also trivial Kan fibrations.

**Proof.**
To show that the map $X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)}$ is a trivial Kan fibration, we must show that every lifting problem every lifting problem

\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d] & X_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] & X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)} } \]

admits a solution, provided that the left vertical map $A \rightarrow A'$ is a monomorphism. Unwinding the definitions, we see that this can be rephrased as a lifting problem

\[ \xymatrix@C =50pt@R=50pt{ (A \star K) \coprod _{ (A \star K_0) } (A' \star K_0) \ar [r] \ar [d] & X \ar [d]^{q} \\ A' \star K \ar [r] \ar@ {-->}[ur] & S. } \]

This problem admits a solution, since the vertical map on the left is a monomorphism (Proposition 4.3.7.1) and $q$ is a trivial Kan fibration.
$\square$