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I think one can give definition of cubical maps of cubical complexes in a similar way to simplicial maps between simplicial sets. Cubical maps between cubical sets is a map of vertices which is compatible with face and degeneracy maps.

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I'll suppress superscripts and subscripts, write $[-,-]$ for $\mathbf{Hom}(-,-)$, and write $i^*$ as $[i,Y]$, etc. The first diagram involves the inclusion $r$ of the horn and a map $a:\Lambda\to [L,X]$, maps $b,c:\Delta\to [K,X],[L,Y]$, and $([i,X],[L,p])$. We have $([i,X],[L,p])a=(b,c)r$, so by the universal property of the pullback (1)\quad ...

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The standard simplicial enrichment on $Simp(C)$ is defined as follows. First, note that since $C$ is cocomplete, it can be considered to be tensored over $Set$: If $X$ is an object of $C$ and $S$ is a set, then $X\otimes S$ is a coproduct of copies of $X$ indexed by elements of the set $S$. If $K$ is a simplicial set and $A$ is an object of $Simp(C)$, we ...

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Here's a very concrete argument. A limit of the cosimplicial object is an $X$ which comes a priori with maps to each of the cosimplicial levels, but since $0$ is weakly initial in $\Delta$, only needs to be given a map $f:X\to \prod FU_i$, from which the other maps are determined. Now $f$ is equalized by the two face maps, so $f$ factors through the ...

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