Crossposted on MathOverflow here.
I'm having trouble with the proof of Lemma 2.9 in "Cohomology of Monoids in Monoidal Categories" by Baues, Jibladze, and Tonks, and I was wondering if someone could clarify a detail. I'll try to summarize the context of the lemma.
Context
Let $(\Bbb A,\circ,I)$ be an abelian monoidal category where $\circ$ is left distributive, i.e. the natural transformation $$(X_1\circ Y)\oplus(X_2\circ Y)\rightarrow (X_1\oplus X_2)\circ Y$$ is an isomorphism. For example, $\Bbb A$ could be the category of linear operads (this is a motivating example of the article). Given an endofunctor $F$ of $\Bbb A$, we define its cross-effect $$F(A|B):=\ker(F(A\oplus B)\rightarrow F(A)\oplus F(B)).$$ The additivization of $F$ is then the functor $F^\text{add}$ defined by $$F^\text{add}(A):=\text{coker}\left(F(A|A)\rightarrow F(A\oplus A)\xrightarrow{F(+)}F(A)\right).$$ The idea is that $F^\text{add}$ is the additive part of $F$.
Let $(M,\mu,\eta)$ be an internal monoid in $\Bbb A$, and let $L_0$ be the endofunctor $L_0:A\mapsto M\circ(M\oplus A)$. Let $L:=L_0^\text{add}$ be the additivization of $L_0$. (In the case of operads, represented as planar trees, I see $L(A)$ as the space of trees whose nodes are all labeled by elements of $M$ except for one leaf, which is labeled by an element of $A$.)
Suppose now that $\Bbb A$ is right compatible with cokernels, i.e. that
for each $A\in\Bbb A$, the additive functor $A\circ-:\Bbb A\rightarrow\Bbb A$ given by $B\mapsto A\circ B$ preserves cokernels.
Then, in the proof of Lemma 2.9, the authors claim the following:
By the assumption that $\Bbb A$ is right compatible with cokernels it follows that $L(L(X))$ is the additivisation of $L_0(L_0(X))$ in $X$ [...].
Remarks
If anyone could provide an explanation of the last claim, I would be very grateful. However, my inability to understand how to show this might be related to two other issues I have:
1) Elsewhere in the literature, cross-effects are only defined when $F$ is reduced, i.e. $F(0)=0$ (e.g. here, section 2). But we can always reduce a functor by taking the cokernel of $F(0)\rightarrow F(X)$, so I don't think it's much of a problem.
2) In the first quote, the authors state that $A\circ -$ is additive, which is quite the opposite of the initial hypothesis that $\circ$ be left distributive, and not necessarily right distributive. How to resolve this apparent conflict?