Variant 11.10.1.7. Let $S$ be a simplicial set. We define full simplicial subcategories
\[ \operatorname{LFib}(S), \operatorname{KFib}(S), \operatorname{RFib}(S) \subset (\operatorname{Set_{\Delta }})_{/S} \]
as follows:
An object of $\operatorname{LFib}(S)$ is a simplicial set $X$ equipped with a left fibration $q: X \rightarrow S$.
An object of $\operatorname{KFib}(S)$ is a simplicial set $X$ equipped with a Kan fibration $q: X \rightarrow S$.
An object of $\operatorname{RFib}(S)$ is a simplicial set $X$ equipped with a right fibration $q: X \rightarrow S$.