This theorem 4.4 from The Kunneth Theorem and the Universal Coefficient Theorem for Kasparov’s Generalized K-functor, Jonathan Rosenberg and Claude Schochet.
My question is, how does naturality of $\gamma$ follow from functoriality of Kasparov product?
Functoriality of Kasparov product: If $f:A\to B$ and $g:B\to C$ are graded $*$-homomorphisms, then $\forall x\in KK(A,B):x\otimes _B[g]=g_*(x)$ and $\forall y\in KK(B,C):[f]\otimes_B y=f^*(y)$.
Another possible approach? Since $K_*(A)=KK_*(\mathbb C,A)$, every element in $KK_*(A,B)$ is sent to a homomorphism $KK_*(\mathbb C,A)\to KK_*(\mathbb C,B)$ by $\gamma$, if this corresponding coincides with Kasparov product, the naturality follows from associativity: $x\otimes_B(y\otimes _C z)=(x\otimes_B y)\otimes_C z$. But I don't know how to show that they coincide.