Comparison of Motivic Homotopy Theories

We construct a comparison functor from the dual category of motivic homotopy category $\mathcal{SH}$ to the category of $\mathbb{A}^1$-invariant localizing motives $\operatorname{Mot}_{\operatorname{loc}}^{\mathbb{A}^1}$ in the sense of Blumberg, Gepner and Tabuada (with $\mathbb{A}^1$-invariance imposed). We as well construct its non-$\mathbb{A}^1$-invariant analogue: a functor from the dual category of Annala-Iwasa-Hoyois's non-$\mathbb{A}^1$-invariant motivic homotopy category $\mathcal{MS}$ to $\operatorname{Mot}_{\operatorname{loc}}$. After the Barr-Beck argument, these functors factor through categories of modules over a dual version of ($\mathbb{A}^1$-invariant) K-theory spectrum $\operatorname{KGL}^{(\mathbb{A}^1)}$. Over a field that admits resolution of singularities, we show that the $\mathbb{A}^1$-invariant factored functor is fully-faithful, while the non-$\mathbb{A}^1$-invariant one is not in general.