My question relates to the proof of the Atiyah-Singer Index Theorem for families of elliptic operators, as presented in "The Index of Elliptic Operators: IV", M. F. Atiyah and I. M. Singer.

Let $A$ be a compact Hausdorff space and $q:A\times \mathbb{C}^n\rightarrow A$ be the projection, then we obtain the induced Thom isomorphism $q_!:K_{\text{cpt}}(A\times \mathbb{C}^n)\rightarrow K(A)$. The map $q_!$ and the analytic index $ind:K_{\text{cpt}}(A\times \mathbb{C}^n)\rightarrow K(A)$ coincide. According to Atiyah this follows from the case $Y$ is a point, and the fact that the analytical index is a homomorphism of $K(Y)$-modules. My question is why are these two properties enough to show the two maps coincide?

Thanks,

Tristan