In this paper, we consider a fractional p-Laplacian system of equations in the entire space RN with doubly critical singular nonlinearities involving a local critical Sobolev term together with a nonlocal Choquard critical term; the problem also includes a homogeneous singular Hardy term; moreover, all the nonlinearities involve singular critical weights. To prove the main result we use a refinement of Sobolev inequality that is related to Morrey space because our problem involves doubly critical exponentss and a version of the Caffarelli-Kohn-Nirenberg inequality. With the help of these results, we provide sufficient conditions under which a weak nontrivial solution to the problem exists via variational methods.