@YosemiteSam I'm still not sure why $P$ must be a sheaf up to shift also it seems so. Could you give more reason?

@pbelmans It seems that you are right. Both morphisms are quasi-compact even if the field extension is not finite.

@LeonidPositselski Thank you very much for your comment. What I want to ask is whether they are dg Morita equivalent instead of quasi-isomorphic. I have edited my question and maybe it makes more sense now.

@JeremyRickard Thank you very much for pointing out the problem in my original question. What I want to ask is whether they are dg Morita equivalent instead of quasi-isomorphic. I have edited my question and maybe it makes more sense now.