Pointwise left fibrations under unstraightening

Lemma 23.2.2. Let \(S\) be a small \(\infty \)-category, and let \(\alpha \colon F\Rightarrow G\) be a natural transformation between functors \(F,G\colon S\to \Cat _{\infty }\). If every functor \(\alpha _s\colon F(s)\to G(s)\) is a left fibration, then the induced functor \[ \Un ^{\cc }(\alpha )\colon \Un ^{\cc }(F)\longrightarrow \Un ^{\cc }(G) \] is a left fibration. Its fiber over \((s,g)\in \Un ^{\cc }(G)\) is the fiber of \(\alpha _s\) over \(g\).