So here is a construction outlined in Higson's note on index theory, pg 46
Let $\mathcal{K}$ denote a $C^*$ algebra of graded compact operators on a graded hilbert space $H=H_0\oplus H_1$. Let $\mathcal{S}=C_0(\Bbb R)$. Let $A$ and $B$ be (not necessarily unital) $C^*$-algebras.
Let $\phi:\mathcal{S} \otimes A \rightarrow B \otimes \mathcal{K}$ be a $*$-algebra homomoprhism.
We want to construction a $K$-theory map $\phi_*:K(A) \rightarrow K(B)$.
He argues the existence as follow, beginning with:
Let $C$ be image of $\mathcal{S} \otimes A$ under $\phi$. Then we obtain homomoprhisms $\phi_{\mathcal{S}}, \phi_A$ of $\mathcal{S}$ and $A$ into the multiplier algebra of $C$.
How does this work? I do not see how this even follows from universal property of multiplier algebras.
Since $\phi\colon \mathcal{S}\otimes A\to C$ is non-degenerate, there exists a unique strictly continuous extension $\tilde \phi\colon M(\mathcal{S}\otimes A)\to M(C)$. Moreover, the universal property of the tensor product gives canonical maps $\iota_{\mathcal{S}}\colon\mathcal{S}\to M(\mathcal{S}\otimes A)$, $\iota_A\colon A\to M(\mathcal{S}\otimes A)$. The $\ast$-homomorphisms $\phi_{\mathcal{S}}$ and $\phi_A$ are given by the obvious compositions of the previous maps.
On a more technical level, the maps $\phi_{\mathcal{S}}$ and $\phi_A$ are given by $$ \phi_{\mathcal{S}}(f)=\lim_\lambda \phi(f\otimes e_\lambda)\\ \phi_A(a)=\lim_j \phi(g_j\otimes a), $$ where $(g_j)$ and $(e_\lambda)$ are approximate units for $\mathcal{S}$ and $A$ respectively and the limits are taken in the strict topology (of course one has to justify that they exist).