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?