Module of type $FP_n$

Question

I'm trying to understand the converse of theorem 1.3 in Robert's Bieri Homological dimension of discrete groups which says that a $\Lambda$-module $A$ is of type $FP_n$ if and only if for every direct product $\Pi \Lambda$ the natural map $Tor_k(\Pi\Lambda,A) \to \Pi Tor_k(\Lambda,A)$ is an isomorphism for $k\in[|1;n-1|]$ and an epimorphism for $k=n$: (iiia) $\implies$ (i) in the book. Don't the $a_i$'s depend on $a$ ? To me the statement is $$(\forall\,a\in A)\,(\exists\,(a_1,\ldots,a_m)\in\Lambda^m)\,(\exists\,(\lambda_1^a,\ldots,\lambda^a_m)\in A^m)\,a=\sum\limits_{i=1}^m\lambda^a_ia_i$$ when we would like to have $$(\exists\,(a_1,\ldots,a_m)\in\Lambda^m)\,(\forall\,a\in A)\,(\exists\,(\lambda_1^a,\ldots,\lambda^a_m)\in A^m)\,a=\sum\limits_{i=1}^m\lambda^a_ia_i$$ Am I missing something?