The composition of two two copulas provides for more sophisticated structures of dependence between variables than many single parameter copula can provide. Further, asymmetrical copulas are readily obtained. Let $\mathbf{A}$ and $\mathbf{B}$ be copulas. Then$$\mathbf{C}_{\alpha,\beta}(u,v) = \mathbf{A}(u^\alpha, v^\beta) \cdot
\mathbf{B}(u^{1-\alpha},v^{1-\beta})\mbox{,}$$
defines a family of copulas $\mathbf{C}_{\alpha,\beta}$, with parameters $\alpha,\beta \in \mathcal{I}:[0,1]$. In particular, if $\alpha = \beta = 1$, then $\mathbf{C}_{1,1} = \mathbf{A}$, and, if $\alpha = \beta = 0$, then $\mathbf{C}_{0,0} = \mathbf{B}$. For $\alpha \ne \beta$, the $\mathbf{C}_{\alpha,\beta}$ is, in general, asymmetric, that is $\mathbf{C}(u,v) \ne \mathbf{C}(v,u)$ for some $(u,v) \in \mathcal{I}^2$.
It is important to stress that copulas $\mathbf{A}_{\Theta_A}$ and $\mathbf{B}_{\Theta_B}$ can be of different families and each parameterized accordingly by the values $\Theta_A$ and $\Theta_B$. This is an interesting observation in the context of building complex copulitic structures in pursuit of fitting asymmetric measures of dependency such as the L-comoments available in the lmomco package. The author does not know whether the copulas $\mathbf{A}$ and $\mathbf{B}$ need be symmetric as the reference makes no stated restriction to that effect. (Symmetry of the copula $\mathbf{C}$ is required for the situation that follows.)
It is possible to simplify the construction of an asymmetric copula for a single copula by the following. Let $\mathbf{C}(u,v)$ by a symmetric copula, $\mathbf{C} \ne \Pi$ (for $\Pi$ see P). A family of asymmetric copulas $\mathbf{C}_{\alpha,\beta}$, with parameters $0 < \alpha,\beta < 1, \alpha \ne \beta$, that includes $\mathbf{C}(u,v)$ as a limiting case, is given by
$$\mathbf{C}_{\alpha,\beta}(u,v) = u^\alpha v^\beta \cdot \mathbf{C}(u^{1-\alpha},v^{1-\beta})\mbox{.}$$
The composite2COP function is based on the more general result given in the former to provide maximum flexibility. For simpler case given in the later, the composite1COP is available.