I am reading the presentation of Cuntz' proof of Bott periodicity for $C^*$-algebras in Wegge-Olsen (Thm. 11.2.1). Here one considers the short exact sequence of $C^*$-algebras $$0 \longrightarrow \mathcal{T}_0 \longrightarrow \mathcal{T} \stackrel{q}{\longrightarrow} \mathbb{C} \longrightarrow 0,$$ where $\mathcal{T}$ is the Toeplitz algebra and $q$ is the homomorphism determined by sending the shift operator $S \in \mathcal{T}$ to $1$. Clearly, we also have the map $j: \mathbb{C} \rightarrow \mathcal{T}$, and $q \circ j = \mathrm{id}_{\mathbb{C}}$.

Now, the main step in the proof is to show that the induced map in $K$-theory $q_*: K_0(\mathcal{T}) \rightarrow K_0(\mathbb{C})$ is an isomorphism.
**But then, it is claimed that it was also clear that also for any $C^*$-algebra $A$, one has that also $$(\mathrm{id}_A \otimes q)_*: K_0(A \otimes \mathcal{T}) \rightarrow K_0(A)$$ is an isomorphism.**

I do understand that since all $C^*$-algebras in the short exact sequence are nuclear, the sequence $$0 \longrightarrow A \otimes \mathcal{T}_0 \longrightarrow A \otimes \mathcal{T} \stackrel{\mathrm{id}_A \otimes q}{\longrightarrow} A \longrightarrow 0$$ is exact for any $C^*$-algebra. But I do not see why knowing that $q_*$ is an isomorphism implies the same thing for $(\mathrm{id}_A \otimes q)_*$. Any suggestions?