On Lurie's Higher Topos Theory, Prop. 2.1.1.3,
Let $F:C \rightarrow D$ be a functor between categories. Then $C$ is cofibered in groupoids over $D$ if and only if the induced map $N(F):N(C)\rightarrow N(D)$ is a left fibration of simplicial sets.
In line 1 of proof, Lurie states $N(F)$ is an inner fibration - how does this follow from Prop. 1.1.2.2?
We only know that $N(C)$ and $N(D)$ as objects has LLP for inner horns.
If you have not already resolve this, I believe this is what Matt means. (Correct me if wrong.)
Consider diagram, $n \ge 2$, $0<i<n$, $S$ a quasicategory, $$ \require{AMScd} \begin{CD} \Lambda^n_i @>h>> S;\\ @VViV @VVfV \\ \Delta^n @>>g> N(C); \end{CD} $$ $g$ is the unique lift of the morphism $i$ with respect to $fh$. (nerves of category have unique lifts)
Given any lift $k:\Delta^n \rightarrow S$ of upper left diagram: $ki=h$ we have $$fki=fh=gi$$ So $fk=g$ from uniqueness. Thus, $f$ is an inner fibration.