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Eilenberg, MacLane: On the groups $H(\Pi,n)$, II: Methods of Computation, Ann. of. Math. 60(1954), No. 1, 49 - 139.
1) Let $a_i \in A_i$ be homogen. $$\nabla(a_1 \otimes \cdots \otimes a_n) = a_1 \nabla a_2 \nabla ... \nabla a_n$$ (well-definied since $\nabla$ is associative)
2) Let $a_i \in (A_i)_m$. $$\Delta(a_1 \otimes \cdots \otimes a_n) = \sum \displaystyle \otimes_{i=1}^n\displaystyle\tilde{d}^{m-j_i}d_0^{j_{i-1}}a_i$$ where the sum is taken over $0 \le j_1 \le \cdots \le j_{n-1} \le m$ and $\tilde{d}^{m-j_0},\;d_0^{j_{-1}}$ \tilde{d}^{m-j_n},\;d_0^{j_0}$has to be interpreted as identity. 1 Yes, there is a generalization to a finite number of simplicial complexes. A reference is Corollary 2.2 in the paper Eilenberg, MacLane: On the groups$H(\Pi,n)$, II: Methods of Computation, Ann. of. Math. 60(1954), No. 1, 49 - 139. Using the definitions from nLab, the maps are given as follows: 1) Let$a_i \in A_i$be homogen. $$\nabla(a_1 \otimes \cdots \otimes a_n) = a_1 \nabla a_2 \nabla ... \nabla a_n$$ (well-definied since$\nabla$is associative) 2) Let$a_i \in (A_i)_m$. $$\Delta(a_1 \otimes \cdots \otimes a_n) = \sum \displaystyle \otimes_{i=1}^n\displaystyle\tilde{d}^{m-j_i}d_0^{j_{i-1}}a_i$$ where the sum is taken over$0 \le j_1 \le \cdots \le j_{n-1} \le m$and$\tilde{d}^{m-j_0},\;d_0^{j_{-1}}\$ has to be interpreted as identity.