History of the characterization of $K$-homology and $\operatorname{KK}$.
Atiyah tried to construct $K$-homology via elliptic differential operators.
Brown-Douglas-Fillmore related $K$-homology to $C^\ast$-algebra extensions.
Kasparov defined $\operatorname{KK}$ via generalised elliptic operators and related it to extensions.
Cuntz described $\operatorname{KK}(A,B)$ as the set of homotopy classes of $\ast$-homomorphisms $qA\to B\otimes \mathcal{K}$ or $q(A\otimes\mathcal{K})\to q(B\otimes \mathcal{K})$ where $qA = \operatorname{Ker}(A\sqcup A \xrightarrow{(\operatorname{id},\operatorname{id})} A)$ ($\sqcup$ is the coproduct), and almost stated its universal property.