Let $\mathcal{A}$ be a monoidal abelian category. Let $A$ and $B$ be simplicial objects in $\mathcal{A}$, and let $N_\ast(-)$ denote the normalized chain complex functor. Let $$AW_{A,B}\colon N_\ast(A\otimes B) \longrightarrow N_\ast(A)\otimes N_\ast(B)$$ and $$ EZ_{A,B}\colon N_\ast(A)\otimes N_\ast(B) \longrightarrow N_\ast(A\otimes B)$$ denote the Alexander-Whitney map and the Eilenberg-Zilber map respectively. Then on homology $AW_{A,B}$ and $EZ_{A,B}$ induce inverse isomorphisms. On the chain level they satisfy $$AW_{A,B}\circ EZ_{A,B} = Id_{N_\ast(A)\otimes N_\ast(B)}$$ and, at least in the case $\mathcal{A}=\mathrm{Ab}$, $$EZ_{A,B}\circ AW_{A,B}\sim Id_{N_\ast(A\otimes B)}.$$ Here $\sim$ denotes chain homotopy.

Is it true that $EZ_{A,B}\circ AW_{A,B}\sim Id_{N_\ast(A\otimes B)}$ for an arbitrary monoidal abelian category $\mathcal{A}?$ If so, does the result appear in the literature, or can a proof be quickly reconstructed from the literature?

**Remark:** What is most important for me is to know (1) how general $\mathcal{A}$ can be in order for the answer to be "yes",
and to know (2) that there is a chain homotopy $EZ_{A,B}\circ AW_{A,B}\sim Id_{N_\ast(A\otimes B)}$, and not just that
$AW_{A,B}$ and $EZ_{A,B}$ induce inverse quasi-isomorphisms.

**Some more information:** In the case $\mathcal{A}=\mathrm{Ab}$, the definition of $AW_{A,B}$ and $EZ_{A,B}$ can be found in Definitions 29.7 of Peter May's book *Simplicial objects in algebraic topology* and on the
nLab page monoidal Dold-Kan correspondence. The definitions are the same for an arbitrary choice of $\mathcal{A}.$
Corollary 29.10 of May's book proves
$AW_{A,B}\circ EZ_{A,B} = Id_{N_\ast(A)\otimes N_\ast(B)}$
and $EZ_{A,B}\circ AW_{A,B}\sim Id_{N_\ast(A\otimes B)}$,
again in the case $\mathcal{A}=\mathrm{Ab}$. Section 8.5 of Weibel's book *An introduction to homological algebra* proves that the Alexander-Whitney and Eilenberg-Zilber maps
are inverse quasi-isomorphisms, this time for an arbitrary abelian category $\mathcal{A}$, and in a more general bisimplicial setting.
§1.2.3 in Lurie's *Higher Algebra*
shows that $AW_{A,B}$ is a quasi-isomorphism in a somewhat more general setting. Are there Alexander-Whitney and shuffle maps for Dold-Kan for abelian categories? is a different mathoverflow question with a similar title to mine.

naturalchain homotopy, i.e., a collection of transformations $H_k\colon N(A\otimes B)_k\to N(A\otimes B)_{k+1}$ which are natural in $(A,B)$ ... $\endgroup$