# Unstable and stable looping and delooping

I have some basic questions on the relation of looping and delooping in the stable and unstable homotopy categories. I state them it in the motivic setting, but if somebody has an answer for an analogue question in the classic setting it would be appreciated.

Let $$X$$ be a scheme, in Voevodsky motivic homotopy theory we have an unstable homotopy category of $$\mathbf{H}(X)$$, a stable homotoy category $$\mathbf{SH}(X)$$ and an infinite suspension $$\Sigma^\infty \colon \mathbf{H}(X) \to \mathbf{SH}(X)$$. We have a looping given by the internal Hom object $$\underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \underline{\phantom{a}}\ )$$ in $$\mathbf{H}(X)$$ and also a looping $$\underline{\mathrm{Hom}}_{\mathbf{SH}(X)}(\mathrm{Th}(\mathcal{O}), \underline{\phantom{a}}\ )=\mathrm{Th}(-\mathcal{O})\wedge \underline{\phantom{a}}\$$ in $$\mathbf{SH}(X)$$. I would like to understand the relation between $$\Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \underline{\phantom{a}}\ )$$ and $$\mathrm{Th}(-\mathcal{O})\wedge \underline{\phantom{a}}\$$ and if for some simple objects they are the same.

To be more precise, let $$L$$ be a line bundle in $$X$$.

1. What is the relation between $$\Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \mathrm{Th}(L))$$ and $$\mathrm{Th}(-\mathcal{O})\wedge \mathrm{Th}(L)$$? And between $$\mathrm{Th}(\mathcal{O})\wedge \Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \mathrm{Th}(L))$$ and $$\mathrm{Th}(L)$$? Are they isomorphic?
2. If they are not isomorphic, it seems to me that they do have the same cohomology. To fix ideas: is their $$K$$-theory $$K_0(\Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \mathrm{Th}(L)))$$ and $$K_0(\mathrm{Th}(L-\mathcal{O}))$$ isomorphic? What about $$K_0(\mathrm{Th}(\mathcal{O})\wedge \Sigma^\infty \underline{\mathrm{Hom}}_{\mathbf{H}(X)}(\mathrm{Th}(\mathcal{O}), \mathrm{Th}(L)))$$ and $$K_0(\mathrm{Th}(L))$$, are they isomorphic?