On the relation between the product of KK-groups and the KK-group of the product

We observe that the canonical map $KK(A,{\prod }_{n\in \mathbb{N}}{B}_n)\to {\prod }_{n\in \mathbb{N}}KK(A,B_n)$ is an isomorphism of abelian groups whenever $A$ enjoys the Universal Coefficient Theorem and $B_n$ are unital, simple and purely infinite C*-algebras. This clarifies an aspect of previous work of Dadarlat–Eilers and Tikuisis–White–Winter.