Let $L\_{Nis}(sPre(Sm\_S))$ be the Nisnevich localization of the category of simplicial presheaves, 
how to see that whether $\mathbb{A}^1$-projections $\mathbb{A}^1\times\_S X\to X$ are closed under homotopy pullback in $L\_{Nis}(sPre(Sm\_S))$? How to compute the homotopy pullback of maps $\mathbb{A}^1\times\_S X\to X$ along itself in $L\_{Nis}(sPre(Sm\_S))$? 



Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization
First, in any model category, weak equivalences are closed under homotopy pullback. That's the whole point of the "homotopy" in "[homotopy pullback](https://ncatlab.org/nlab/show/homotopy+pullback)." 


In your question, you are missing a step. [Generally](https://arxiv.org/pdf/1605.00929.pdf), $Sm\_S$ is the category of smooth and finite type schemes over a scheme S of finite dimension. This category doesn't have good categorical behavior (it's not bicomplete, for example), so Voevodsky embedded it into simplicial presheaves $SPre(Sm\_S)$, i.e. functors into simplicial sets. This has a model structure and you can do the localization $L\_{Nis} SPre(Sm\_S)$ that you wanted. To get the statement about $\mathbb{A}^1$-weak equivalences, you do a further localization to make $\mathbb{A}^1$ contractible, and it's this model structure that tells you homotopy pullbacks preserve weak equivalences. See [Morel-Voevodsky](http://www.math.ias.edu/vladimir/files/A1_homotopy_with_Morel_published.pdf).


The resulting model category is [proper](https://ncatlab.org/nlab/show/proper+model+category), meaning that if you look at a diagram $A \to B \gets C$ where one of those maps is a fibration, then you can compute homotopy pullback as just the regular old pullback, which is easy in presheaf categories. For a proof, see [Jardine](https://www.math.uni-bielefeld.de/documenta/vol-05/15.pdf). If neither of those maps is a fibration, you can do fibrant replacement to replace one by a fibration.


