Remark 5.3.0.2. After restricting to diagrams of Kan complexes, the results of this section supply a dictionary
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Left fibrations $\operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$} \} \ar@ <.8ex>[d]^-{ \operatorname{sTr}_{(-)/\operatorname{\mathcal{C}}}} \\ \{ \textnormal{Functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$} \} \ar@ <.8ex>[u]^-{ \operatorname{N}_{\bullet }^{(-)}(\operatorname{\mathcal{C}}) } } \]
This dictionary was formulated in work of Heuts and Moerdijk (using the language of model categories) which is closely related to the contents of this section. For more details, we refer the reader to [MR3318247].