Cubical cup poduct with a unit 1-cochain and the coboundary operator

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CoreSpace
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TL;DR
equivalence between cup product and coboundary operator
Define a u-coboundary operator u-d on a cubic p-cochain f as cup product multiplication U from the left with a unit 1-cochain: (u-d)f = 1Uf.
Because 1U1=0 and associativity of U, we have (u-d)^2=0.
What is the relation of u-d and the standard coboundary operator d?
Are they the same?