$\require{AMScd}$One thing that needs to be checked to give an interpretation of type theory in simplicial sets (as in [Kapulkin-Lumsdaine](https://arxiv.org/abs/1211.2851)) is that "the base of the universal fibration is fibrant". Expanding the definitions reduces this to the following statement.

> 
> $(\*)$ Given a Kan fibration $p : X \to A$ and an acyclic cofibration $i : A \to B$, there exists a square
>  $$
> \begin{CD}
>  X @>j>> Y \\
>  @V p VV @VV q V \\
>  A @>>i> B
> \end{CD}
> $$
>  in which $q$ is a Kan fibration and the square is a pullback square.
> 
> 
> 

It suffices to handle the case when $i : A \to B$ is one of the generating acyclic cofibrations $\Lambda^n\_k \to \Delta^n$.

I have seen only one proof of $(\*)$, which relies on the theory of minimal fibrations. I'm wondering about alternative approaches, so my main question is

> 
> Are there proofs of $(\*)$ which do not use minimal fibrations?
> 
> 
> 

In particular, a probably related question is whether, for fixed $i : A \to B$, the construction in $(\*)$ can be carried out functorially in $X$ in an appropriate sense.

Observe that in a square such as in $(\*)$, the map $j$ is automatically an acyclic cofibration (it is a monomorphism, and a weak equivalence because $\mathrm{SSet}$ is right proper). So $X \xrightarrow{j} Y \xrightarrow{q} B$ is an acyclic cofibration-fibration factorization of $ip : X \to B$, but with the added constraint that $j : X \to Y$ can only attach simplices which live above $B \setminus A$. Perhaps it is possible to modify a known method for constructing factorizations so that it adheres to this additional constraint?