Ultimately, I'm trying to figure out whether or not the full subcategory in $\mathbf{sSet}$ spanned by Kan complexes is finitely complete (as a $1$-category).
Since fibrations are stable under pullback in general, I know that Kan complexes are closed under finite products, so the question boils down to whether the pullback in the square
$\require{AMScd}$
\begin{CD}
K\times\_LK' @>>> K\\
@VVV @VVV\\
K' @>>> L
\end{CD}
where $K$, $K'$, and $L$ are all Kan complexes must have $K\times\_LK'$ as a Kan complex also.
In my limited experience, I feel like this isn't true since it's not true in a general model category, but I can't construct a counterexample.


I'm honestly pretty bad at creating Kan complexes in general, and my usual go-to's (simplicial groups and nerves of groupoids) are actually preserved under taking pullbacks (the former because $\mathbf{Grp}$ is complete and limits of simplicial sets/groups are computed levelwise; the latter because the nerve is fully faithful from $\mathbf{Cat}$ to $\mathbf{sSet}$ and pullbacks of groupoids are groupoids). Maybe my intuition is wrong?



($1$-)pullbacks of Kan complexes
Take any simplicial set $X$ which is not a Kan complex. Let $K$ be a Kan replacement of $X$, and let $L$ be a Kan replacement of the pushout $K\amalg\_X K$. Then the two maps $K\to L$ are levelwise injective, and the pullback $K\times\_L K$ is precisely $X$.


