An isomorphism in the proof of Bott periodicity

Question

In Efton Park's book on topological $K$-theory, to prove Bott periodicity, he considers the space $\mathcal{S}'X=(X\times S^{1})/(X\times\{1\})$ for a compact Hausdorff space $X$, and proves that there is a natural isomorphism $K^{0}(X)\rightarrow K^{-1}(\mathcal{S}'X)$. Is it also true that $K^{-1}(X)\cong K^{0}(\mathcal{S}'X)$?