This is a basic question about the meaning of functoriality of cup products.
On page 105 of "Cohomology of Groups" appearing in Cassels and Frohlich's Algebraic Number Theory, Atiyah and Wall define a certain family of homomorphisms $$\hat{H}^p(G,A)\otimes_\mathbb Z \hat{H}^q(G,B) \to \hat{H}^{p+q}(G,A\otimes_\mathbb Z B)$$ where $p$ and $q$ are arbitrary integers, $G$ is a finite group, $A$ and $B$ are $G$-modules and $\hat{H}^p(\;,\;)$ are the Tate cohomology groups.
They then claim that these homomorphisms are functorial in $A$ and $B$.
Could anyone explain what this means exactly? Do we consider both sides of the homomorphisms above as bifunctors in $(A,B)$ and then show that the homomorphisms preserve functoriality?
Many thanks.