∀ {X' X : GrpCat} {Y : ProfiniteGrp.{u_1}} (f : X' ⟶ X)
  (g : X ⟶ (CategoryTheory.forget₂ ProfiniteGrp.{u_1} GrpCat).obj Y),
  (ProfiniteGrp.ProfiniteCompletion.homEquiv X' Y).symm (CategoryTheory.CategoryStruct.comp f g) =
    CategoryTheory.CategoryStruct.comp (ProfiniteGrp.profiniteCompletion.map f)
      ((ProfiniteGrp.ProfiniteCompletion.homEquiv X Y).symm g)