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If you are looking for a reference, Propostion 3.3.18Propostion 3.3.18 of Munson Volic's "Cubical homotopy theory" (available at http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf) is doing what you want.

If you are looking for a reference, Propostion 3.3.18 of Munson Volic's "Cubical homotopy theory" (available at http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf) is doing what you want.

If you are looking for a reference, Propostion 3.3.18 of Munson Volic's "Cubical homotopy theory" is doing what you want.

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Geoffroy Horel
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If you are looking for a reference, Propostion 3.3.18 of Munson Volic's "Cubical homotopy theory" (available at http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf) is doing what you want.