Let $A,B,C$ be $C^\ast$-algebras. Suppose $B$ and $C$ to be strongly morita equivalent. Then $KK(A,B)\cong KK(A,C)$.
Could someone provide a reference or proof of this fact?
I guess the imprimitivity bimodule defines an element in $KK(B,C)$ and Kasparov product with this element is the desired isomorphism, but I don't know how to fill the details. Also I am unsure whether I should add $\sigma$-unitality/separability as hypothesis.
In Blackadar's K-Theory for Operator Algebras (first edition), paragraph 13.7.1 is an exercises that guides you to prove that if $A$ and $B$ are strongly Morita equivalent $\sigma$-unital C*-algebras, then $A$ and $B$ are stably isomorphic. Hence use the fact that KK-Theory is stable in both arguments to get to the sought isomorphism.