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Suppose I have a 3-Copula which I would like to construct with two 2-Copula's, as the following construct:

$$ C_2(u, C_1(v,w)) = C(u,v,w) $$

My question is, if $C$ is known to be a valid copula (i.e. follows the properties such as groundedness, etc ...), then are $C_1$ and $C_2$ unique? I think this is the reverse way of looking at the copula compatibility problem, but I am not 100% sure.

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