Remark 11.10.1.9 (Functoriality). Let $f: S' \rightarrow S$ be a morphism of simplicial sets. Then $f$ determines a functor of simplicial categories $f^{\ast }: (\operatorname{Set_{\Delta }})_{/S} \rightarrow (\operatorname{Set_{\Delta }})_{/S'}$, given on objects by the formula $f^{\ast }(X) = S' \times _{S} X$. Since the collection of cartesian fibrations is stable under base change (Remark 5.1.4.6), $f^{\ast }$ restricts to a simplicial functor $\operatorname{Cart}(S) \rightarrow \operatorname{Cart}(S')$, which we will also denote by $f^{\ast }$. Similarly we obtain pullback functors
\[ f^{\ast }: \operatorname{CCart}(S) \rightarrow \operatorname{CCart}(S') \quad \quad f^{\ast }: \operatorname{KFib}(S) \rightarrow \operatorname{KFib}(S') \]
\[ f^{\ast }: \operatorname{LFib}(S) \rightarrow \operatorname{LFib}(S') \quad \quad f^{\ast }: \operatorname{RFib}(S) \rightarrow \operatorname{RFib}(S'). \]