Let $\Lambda^n\_i \subseteq \Delta^n$ be an inner horn, and let $X \rightarrow \Lambda^n\_i$ be a left fibration. Does there exist a left fibration $Y \rightarrow \Delta^n$ such that $X = Y \times\_{\Delta^n} \Lambda^n\_i$?


The only results in that direction that I am aware of are the following two lemmas from [*Left fibrations and homotopy colimits*](http://arxiv.org/abs/1308.0704) by Heuts, Moerdijk:



> 
> **Lemma 7.3.** Consider a pullback square of simplicial sets 
>  $\require{AMScd}$
>  \begin{CD}
>  X \times\_Y Z @>g>> Z\\
>  @VVV @VV p V\\
>  X @>>f> Y
> \end{CD}
>  in which $f$ is inner anodyne and $p$ is a left fibration. Then $g$ is a trivial cofibration in the Joyal model structure.
> 
> 
> **Lemma 7.4.** Let $0 < k < n$ and let $p : A \rightarrow \Lambda^n\_k$ be a left fibration. Then there exists a left fibration $q : B \rightarrow \Delta^n$ and an equivalence
>  \begin{CD}
>  A @>g>> \Lambda^n\_k \times\_{\Delta^n} B\\
>  @VpVV @VVV\\
>  \Lambda^n\_k @= \Lambda^n\_k
> \end{CD}
>  in the covariant model structure over $\Lambda^n\_k$.
> 
> 
> 



Extending a left fibration along an inner horn
Yes. This is shown using minimal left fibrations in Cisinski's book [Higher categories and homotopical algebra](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf), see the proof of Theorem 5.2.10 therein. This theorem states that the simplicial set $\mathscr{S}$, whose $n$-simplices are (essentially) the left fibrations over $\Delta^n$ with small fibres, is a quasi-category.


