It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist compatible proper model structures for the stable homotopy categories of motivic $S^1$-spectra and $T$-spectra, so that the natural connecting functors (between all the four categories) are left Quillen ones. Could you suggest me a reference for this fact? I do not need tensor structures for motivic spectra; I would prefer to present $T$ as $A^1/G_m$. Besides, I need simplicial sheaves only because for embeddings $X\to Y\to Z$ of smooth varieties, $j\ge 0$, I want $(Y/X)\wedge T^j\to (Z/X)\wedge T^j\to (Z/X)\wedge T^j$ to be a cofibration sequence (in the stable $S^1$-homotopy category); does there exist a reference for this fact? Also, where could I find some more or less canonical notation for these functors and for their adjoints (I also need the functor from $T$-spectra to Voevodsky's motives and its adjoint, though only on the homotopy level)? I have looked at several sources on motivic homotopy theory (including some papers of Morel and Jardine); yet I was not able to find a single reference for all these things.