I have a question about the proof of lemma 6.4.12 in the book Algebraic Operads (Loday-Vallette) which I do not seem to be able to fully complete on my own. Hopefully, somebody here can point out what I am not seeing.

Let me sketch the situation. Given a cooperad $(C,\Delta,\epsilon)$ and a operad $(P,\gamma,\eta)$, we consider its free right P-module $C \circ P$. Let $\alpha: C \longrightarrow P$ be a twisting morphism (of degree $-1$). The map $d^r_\alpha$ is then defined as follows $$ C \circ P \overset{\Delta_{(1)} \circ P}{\longrightarrow} (C \circ_{(1)} C) \circ P \overset{(C \circ_{(1)} \alpha) \circ P}{\longrightarrow} (C \circ_{(1)} P) \circ P \cong C \circ (P ; P\circ P) \overset{C \circ (P;\gamma)}{\longrightarrow} C \circ (P ; P) \cong C \circ P$$ In concrete terms, $$d^r_\alpha( c ; p_1,\ldots,p_n) = \sum_i \pm (c_{(1)}; p_1,\ldots, \alpha c_{(2)} \circ (p_i,\ldots,p_j), \ldots,p_n) $$ where $\Delta_{(1)}(c) = c_{(1)} \otimes (i, c_{(2)})$ denotes the sweedler notation.

The proof of the lemma then states the following $$ [ d^r_\alpha , d^r_\alpha ] = d^r_{[\alpha,\alpha]}$$ which reduces to $$ (d^r_\alpha)^2 = d^r_{\alpha * \alpha}$$ The book says the computation is similar to the case of twisted complex for associative (co)algebras. However, I think extra terms appear on the left-hand side, namely $$ (c_{(11)}; p_1, \ldots, \alpha c_{(12)} \circ (p_i,\ldots,p_j), \ldots ,\alpha c_{(2)} \circ (p_k,\ldots,p_l), \ldots ,p_n)$$ and $$(c_{(11)}; p_1, \ldots, \alpha c_{(2)} \circ (p_i,\ldots,p_j), \ldots ,\alpha c_{(12)} \circ (p_k,\ldots,p_l), \ldots ,p_n) $$ Even using coassociativity, I still do not see how these terms cancel as they are not of the right shape? Perhaps I am overlooking something simple?

Thanks in advance for taking your time to read and help!