For a given Copula what distribution is the resulting joint density?

Question

Say you have 2 random variables $x, y$ and a copula $C$ to model their interdependence. The two distributions are made uniform and the copula has some form (e.g., Gaussian, Clayton).

Given that, does that mean that the resulting joint cumulative distribution function $F(x,y)=C(F(X),F(Y))$ has the same form as the copula (e.g., Gaussian, Clayton).

Answer 1

If you are referring to the structure of the two mathematical expressions, $F(x,y)$ and $C(F(x), F(y))$ the answer is "no", at least "not in general".

So indeed, we have here two different functions that give exactly the same values, when one is fed with $(x,y)$ and the other is fed with $(F(x), F(y))$, $\forall \,(x,y)$ in the joint support.