If $X$ is a Kan complex, then it is an easy consequence of the existence of the Quillen model structure or of the basic theory of anodyne extensions, that $X^{K}$ is also Kan. However, I am interested in a "direct" proof of this fact, appealing to neither of the above (which is apparently how these sorts of things were proved before Gabriel-Zisman). 


Here is the argument I would like to complete: 


Let $X$ be a Kan complex, and let $K$ be any simplicial set. It suffices to show that $Hom(K\times \Delta^{n},X) \rightarrow Hom(K \times \Lambda\_{k}^{n},X)$ is surjective for all horn inclusions. Since $K\simeq \text{colim}\_{m} sk\_{m}(K)$ is a filtered colimit, the claim follows if 
$$Hom(sk\_{m}(K)\times \Delta^{n},X) \rightarrow Hom(sk\_{m}(K) \times \Lambda\_{k}^{n},X)$$ is surjective for all $m \ge 0$. Proceed by induction. The base case is easy, but the induction step is what's giving me trouble. By the induction hypothesis, we can choose a solution


$$ h: sk\_{m-1}K \times \Delta^{n} \rightarrow X$$


to the lifting problem


$\require{AMScd}$
$\begin{CD}
sk\_{m-1}K \times \Lambda^{n}\_{k} @>>> sk\_{m}K \times \Lambda^{n}\_{k} @>f>> X \\
@VVV @VVV \\
sk\_{m-1}K \times \Delta^{n} @>>> sk\_{m}K \times \Delta^{n} \\
\end{CD}$


However, as the square above is not a pushout, we can't immediately get a solution to the lifting problem of interest. Now my thought is that maybe there's some way to directly construct the desired lift $g:sk\_{m}K \times \Delta^{n} \rightarrow X$ from $f$ and $g$, since all we need to do is specify the images of the nondegenerate simplices of $sk\_{m}K \times \Delta^{n}$ which are in neither $sk\_{m-1}K \times \Delta^{n}$ nor $sk\_{m}K \times \Lambda\_{k}^{n}$. However, this is where I run into a wall, as I don't know a simple way to describe these.


Ideally, I would like to see this induction step completed. Does anyone have a solution to this or suggestions for possible solutions? It would be just as well if someone knows a different but equally "direct" argument, not appealing to the machinery mentioned above. 



$X^K$ a Kan complex, without model structure or anodyne extensions
For me, the 'usual' way is as suggested by მამუკა ჯიბლაძე. This is based on a systematic analysis of the simplices of $\Delta^m\times \Delta^n$ and of those missing in the corresponding subcomplex involving the k-horn of $\delta^n$. This sort of thing is classically well known and is based on collapsing arguments from `combinatorial' topology. It is handled in great detail in Peter May's Simplicial Objects in Algebraic Topology, starting on page 16.


