Consider two random variables $\epsilon_1, \epsilon_2$ with marginal cdf respectively $F_1, F_2$ and joint cdf $F$.
Is it true that for any $F, F_1, F_2$ we can write $$ F(\epsilon_1, \epsilon_2)=C(F_1(\epsilon_1), F_2(\epsilon_2); \rho) $$ where $C$ is a bivariate copula function with parameter $\rho$ controlling for the correlation between the two random variables?
If not, could you provide a counter-example?