Suppose $f : X \to Y$ is a morphism of simplicial sets which is degreewise surjective and where the fiber over any vertex of $Y$ is an acyclic kan complex. Is $f$ necessarily an acyclic kan fibration? Asking just for surjectivity on vertices isn't strong enough because of eg $\Delta[0]\sqcup \Delta[0] \to \Delta[1]$. I sort of doubt this is true, but I can't think of a counterexample.
I'm also sort of wondering whether there's a way to make precise the idea that an (acyclic) kan fibration $X \to Y$ is a family of (contractible) kan complexes over $Y$ varying in a good way, eg by coming up with some nice description of them as objects of $s\mathsf{Set}/Y \simeq \operatorname{Psh}(Y)$. Of course you can just translate the lifting condition into a statement about representables and horns mapping into $X$ (in the slice category) but I'd prefer something that more directly says "relative kan complex"
No. Take the cone $C$ of any contractible Kan complex $K$. $C$ projects onto $Δ^1$ via the canonical map. Pick any two vertices $x$ and $y$ of $K$ not connected by an edge. Then the cone on $x$ and $y$ is a 2-horn in $C$ that does not have a filler, but its projection to $K$ does have one.