Suppose $f(x, y)$ is a two-variable probability density function. Let $f_x(x)$ be a peripheral density function defined by $f_x(x) := \int_{-\infty}^{\infty} f(x, y) \, dy$. In the similar vein, we have $f_y(y) := \int_{-\infty}^{\infty} f(x, y) \, dx$. The copula is defined as follows$\colon$
$$ C(u, v) := f\left(f_x^{-1}(u), f_y^{-1}(v)\right). $$
Q. What is a heuristic explanation of $C(u, v)$ with respect to the probability theory in view of the original $f(x, y)$? How is $C(u, v)$ related to $f(x, y)$ from the probability reasoning?
I would like to know very simple conceptual explanation, or a simple example.
The copula should be defined in this way $C(u,v)=F(F_x^{-1}(u),F_y^{-1}(u))$
...see my comments for further clarifications.