It doesn't look like Washington shows that $\widehat{A\times B}\cong\widehat A\times\widehat B$, but the fact is certainly mentioned in the proof of Lemma 3.1 ($G\cong\widehat G$ noncanonically). Many thanks.

Proposition 3.4 itself says that $(H^\perp)^\perp=H$, and the preceding setup is that $H$ is a subgroup of a finite abelian group $G$. $H^\perp$ is defined to be $\{\chi\in\widehat G:\chi(h)=1,\forall h\in H\}$, so I think this matches up with your comment on the question.

Thanks @KConrad, that's very helpful. If I understand correctly, I believe you are referring to Proposition 3.4 in Introduction to Cyclotomic Fields.