Say I have a set of combinatorial Dyson-Schwinger equations of the form
$$\begin{align} X_1 &= \mathbb{1} + \alpha B_+^a (f_1(X_1,...X_N)) \\ & ... \tag{1} \\ X_N &= \mathbb{1} + \alpha B_+^n (f_1(X_1,...X_N)) \end{align}$$
that has an unique solution, solved by using the ansatz
$$X_j = \mathbb{1} + \sum_k^\infty c_{j,k} \alpha^k \tag{2}$$
How can I check if the elements $c_{j,k}$ form a Hopf subalgebra, by use of the coproduct? The coproduct is defined by
$$\Delta \circ B^i_+ = B^i_+ \otimes \mathbb{1} + (\text{id} \otimes B^i_+) \Delta \tag{3}$$
Thank you in advance.
