Remark 11.10.1.8. Let $q: X \rightarrow S$ be a cocartesian fibration of simplicial sets and let $q': X' \rightarrow S$ be a left fibration. Then $q'$ is also a cocartesian fibration, and every edge of $X'$ is $q'$-cocartesian (Proposition 5.1.4.15). In particular:
Every morphism $f: X \rightarrow X'$ in $(\operatorname{Set_{\Delta }})_{/S}$ carries $q$-cocartesian edges of $X$ to $q'$-cocartesian edges of $X'$. That is, we have $\operatorname{Fun}_{/S}^{\operatorname{CCart}}(X,X') = \operatorname{Fun}_{/S}(X,X')$.
The simplicial set $\operatorname{Fun}_{/S}(X,X')$ is a Kan complex (Corollary 4.4.2.5), and is therefore equal to its core $\operatorname{Fun}_{/S}(X,X')^{\simeq }$.
It follows that we can regard $\operatorname{LFib}(S)$ as a full subcategory of the simplicial category $\operatorname{CCart}(S)$. Similarly, we can regard $\operatorname{RFib}(S)$ as a full subcategory of the simplicial category $\operatorname{Cart}(S)$. We therefore obtain a diagram of fully faithful inclusions