Thank you Theo for the corrections in the source text. It is my first time on MO and I am still trying to understand what it's possible to do. I do not yet understand why some latex instructions work well at a moment and not later.
About your answer : I don't think that it is exactly what I want, since I consider my hypercubes in an (additive) category C. So, if I understand well your suggestion, I should embed the diagram $X overset{f}rightarrow Y overset{g}rightarrow Z$ into the diagram
$begin{array}
X & overset{f}rightarrow & Y \
downarrow && downarrow \
Y & overset{g}rightarrow & Z
end{array}$
where the vertical arrows are 0. This square is commutative, but where is the composition $g circ f$? Even if you consider the identity from Y on the top to Y on the bottom, the diagram wouldn't be commutative.
I hope it is clear that I think to all the diagrams, both `I' and hypercubes, as living in an additive category C, and in the embedding I want to preserve compositions and commutativity (indeed, it gives a functor which is a presheaf from I to C).
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