We give an equivalence of categories between certain subcategories of modules of pro-p Iwahori-Hecke algebras and modulo p representations.2010 Mathematics Subject Classification. 22E50,20C08.Let J be a subset of ∆ and denote the corresponding standard parabolic subgroup by P J . Let L J be the Levi part of P J containing Z. Then K ∩L J is a special parahoric subgroup and I(1) J = I(1)∩L J a pro-p Iwahori subgroup. Attached to these, we have many objects. For such objects we add a suffix J, for example, the pro-p Iwahori-Hecke algebra attached to (L J , I(1) J ) is denoted by H J . There are two exceptions: base T w and E(w) for H J is denoted by T J w and E J (w), respectively. For each J ⊂ ∆, we have twoAn element E(λ) belongs toThe second one follows from the following fact and [Abe16, Lemma 2.6]: a basis of H − J ∩ A J is given by {E(λ)} where λ runs through as above [Abe, Lemma 4.2].) Since w containsLemma 3.12. We regard A w J as a subalgebra of A J via the above embedding. Then n w J M is uniquely extended to A J , namely there exists a unique A J -module M J such that supp M ) are both isomorphisms.Proof. The first one is isomorphism by the similar argument in the proof of Lemma 3.11. Take λ 0 ∈ Λ(1) such thatSince E J (λ 0 ) is invertible in A J , it is also invertible on n w J M J . (Note that n w J M J is an A J -module.) Hence the second homomorphism is an isomorphism.Therefore we get