Let $C(u_1,u_2;\theta)$ be a bivariate parametric copula.
I know if $C(u_1,u_2;\theta)$ is the Gaussian copula, then $\partial_{\theta} C(u_1,u_2;\theta)>0$ for any $u_1$ and $u_2$ (it follows from the property of bivariate standard normal CDF). I also plot the Frank copula in the statistic software and the graph suggests that the Frank copula is also monotonic in $\theta$.
I wonder whether there is any general result on the monotonicity of any bivariate parametric copula $C(u_1,u_2;\theta)$ w.r.t. $\theta$ (or at least $\theta\mapsto C(u_1,u_2;\theta)$ is a one-to-one mapping for any $u_1$ and $u_2$, since otherwise $\theta$ will not be uniquely identified from the observable joint distribution).
Can someone maybe point to some results or references in this respect?