The $Spin^{\mathbb{C}}$-structure is well-known to be defined as the principal $Spin^{\mathbb{C}}$-bundle that covers the oriented orthonormal frame bundle of an oreinted Riemannian manifold.
I've also read definition saying that the $Spin^{\mathbb{C}}$-structure is a circle bundle over the total space of oriented orthonormal frame bundle, which is also equipped with a Principal $Spin^{\mathbb{C}}$-bundle structure.
My question is, defining this way, how could we have the fiber is $Spin^{\mathbb{C}}$? Shouldn't the fiber just be $SO(n)\times U(1)$, which is clearly not $Spin^{\mathbb{C}}$ since we have the identification that $Spin^{\mathbb{C}}/\mathbb{Z}_2 \simeq SO(n) \times U(1)$ ?
Any comment is appreciated. I've attached the screenshot below for the exact sentence I encountered in the text. 