In proving the the existence of the Reedy model structure on the category of simplicial objects in a model category $\mathcal{C}$, Goerss-Jardine prove there is a pullback square induced by a map of simplicial objects $f:X\to Y$, namely 
$$\begin{array}{ccc}
Y\_n\times\_{M\_{n,k+1}}M\_{n,k+1}X& \to & X\_{n-1}\\
\downarrow & & \downarrow\\
Y\_n\times\_{M\_{n,k}Y}M\_{n,k}X& \to & Y\_n\times\_{M\_{n-1,k}Y}M\_{n-1,k}X
\end{array}$$
and I am having some trouble seeing exactly how they derive this pullback. For reference, this is lemma VII.2.2 in Goerss-Jardine. 


Here is what I have attempted so far. Here, Goerss-Jardine define
$$
M\_{n,k}X = M\_{\Delta^{n,k}}X
$$
where $\Delta^{n,k}$ is the simplicial set $d^0\Delta^{n-1}\cup \cdots \cup d^k\Delta^{n-1}$ and for a simplicial set $K$ and a simplicial object $X$ in $\mathcal{C}$, you can take
$$
M\_KX:= \hom^{\Delta^\mathrm{op}}(K,X)
$$
From a standard pushout
$$\begin{array}{ccc}
\Delta^{n-1,k}& \to & \Delta^{n-1}\\
\downarrow & & \downarrow\\
\Delta^{n,k}& \to & \Delta^{n,k+1}
\end{array}$$
Since $M\_{(-)}X$ takes pushouts to pullbacks, we get a pullback 
$$\begin{array}{ccc}
M\_{n,k+1}X& \to & X\_{n-1}\\
\downarrow & & \downarrow\\
M\_{n,k}X& \to & M\_{n-1,k}X
\end{array}$$


and we get a similar pullback for the partial matching objects on $Y$. My attempt at getting the desired pullback was to enlarge the diagram to the following 3-by-3 diagram
$$
\begin{array}{ccccc}
Y\_n & \to & Y\_{n-1} & \leftarrow & X\_{n-1} \\
\downarrow & & \downarrow & & \downarrow \\
M\_{n,k}Y & \to & M\_{n-1,k}Y & \leftarrow & X\_{n-1}\\
\uparrow & & \uparrow & & \uparrow \\
M\_{n,k}X & \to & M\_{n-1, k}X & \leftarrow & X\_{n-1}
\end{array}
$$
The desired pullback diagram is then supposed to be obtained from this 3-by-3 diagram by taking the horizontal limits and then taking the corresponding pullback. I tried commuting these two limits, so I thought of taking the limits of the row and then taking the corresponding pullback, but this didn't really seem to simplify the problem. 


I am not sure if my attempt can be made to work and I am just not seeing something, or perhaps I am unaware of some category theory trick that Goerss-Jardine are using. Anyway, any hints would be greatly appreciated. 



pullback square in Goerss-Jardine
I would understand this proof as describing the limit of a "deleted 3-cube" in two different ways. 


We have a square involving maps $X\_n\to X\_{n-1}$, $X\_n\to M\_{n,k}X$, $X\_{n-1}\to M\_{n-1,k}X$, and $M\_{n,k}X\to M\_{n-1,k}X$; this square maps to a similar square with $X$ replaced with $Y$. Altogether it is a cube. I can't draw this cube here.


Now let's completely forget about the "maximal vertex" $X\_n$. The pullback you want to compute is actually the inverse limit of the *deleted* cube. (It is also equal to the limit of your 3-by-3 diagram.)


When you compute the limit of a deleted 3-cube, you can slice it up in several different ways, so that it is equal to a pullback of three objects, one of which is a vertex, and the other two are pullbacks. Goerss-Jardine have set it up one way. But we can also slice it up as the limit of 
$$ Y\_n \to Y\_{n-1}\times\_{M\_{n-1,k}Y}M\_{n,k}Y \leftarrow X\_{n-1}\times\_{M\_{n-1,k}X} M\_{n,k}X.
$$
GJ identify the rightmost term as $M\_{n,k+1}X$. The same argument shows that the middle term is equal to $M\_{n,k+1}Y$.


