Let $A$ be an abelian group with $G$ action, then I call the tate map $$(-)^{tG}: Ab^G \rightarrow Ab$$ $$ A \mapsto \operatorname{coker}(Nm:A_G \rightarrow A^G)$$
where $Nm:x \mapsto \sum_g gx$, is the norm map.
I heard that this map is lax monoidal - but I don't see how. In fact I don't even see how there is a well defined map $$ A^{tG} \times B^{tG} \rightarrow (A \otimes_{\Bbb Z} B)^{tG}$$