Remark 5.1.3.6. In the situation of Proposition 5.1.3.5, suppose that $q_{Y}$ is a left fibration and the inclusion $X' \hookrightarrow X$ is left anodyne. Then the restriction map $\theta : \operatorname{Fun}_{/S}(X, Y) \rightarrow \operatorname{Fun}_{/S}(X', Y)$ is a trivial Kan fibration (this follows immediately from Proposition 4.2.3.4). Similarly, if $q_{Y}$ is a right fibration and the inclusion $X' \hookrightarrow X$ is right anodyne, then $\theta $ is a trivial Kan fibration.

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