-- Module: Mathlib.CategoryTheory.Sites.Continuous
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
E : Type u₃
inst_2 : Category.{v₃, u₃} E
F : Functor.{v₁, v₂, u₁, u₂} C D
G : Functor.{v₂, v₁, u₂, u₁} D C
adj : F ⊣ G
J : GrothendieckTopology.{v₁, u₁} C
K : GrothendieckTopology.{v₂, u₂} D
inst_3 : F.IsContinuous J K
inst_4 : G.IsContinuous K J
P : Sheaf.{v₂, v₃, u₂, u₃} K E
h :
  InducedCategory.Hom.hom
      (CategoryStruct.comp (Y :=
        Functor.obj (F.sheafPushforwardContinuous E J K)
          (Functor.obj (G.sheafPushforwardContinuous E K J) (Functor.obj (F.sheafPushforwardContinuous E J K) P)))
        (Functor.map (F.sheafPushforwardContinuous E J K)
            (NatTrans.app
              {
                app := fun (P : Sheaf.{v₂, v₃, u₂, u₃} K E) ↦
                  { hom := NatTrans.app (unit (Adjunction.whiskerLeft E adj.op)) (P.obj (P := Presheaf.IsSheaf K)) },
                naturality := ⋯ }
              P) :
          Functor.obj (F.sheafPushforwardContinuous E J K) P ⟶
            Functor.obj (F.sheafPushforwardContinuous E J K)
              (Functor.obj (G.sheafPushforwardContinuous E K J) (Functor.obj (F.sheafPushforwardContinuous E J K) P)))
        (NatTrans.app
            {
              app := fun (P : Sheaf.{v₁, v₃, u₁, u₃} J E) ↦
                { hom := NatTrans.app (counit (Adjunction.whiskerLeft E adj.op)) (P.obj (P := Presheaf.IsSheaf J)) },
              naturality := ⋯ }
            (Functor.obj (F.sheafPushforwardContinuous E J K) P) :
          Functor.obj (G.sheafPushforwardContinuous E K J ⋙ F.sheafPushforwardContinuous E J K)
              (Functor.obj (F.sheafPushforwardContinuous E J K) P) ⟶
            Functor.obj (𝟭 (Sheaf.{v₁, v₃, u₁, u₃} J E)) (Functor.obj (F.sheafPushforwardContinuous E J K) P))) =
    InducedCategory.Hom.hom (𝟙 (Functor.obj (F.sheafPushforwardContinuous E J K) P))
⊢ CategoryStruct.comp (Y :=
      Functor.obj (F.sheafPushforwardContinuous E J K)
        (Functor.obj (G.sheafPushforwardContinuous E K J) (Functor.obj (F.sheafPushforwardContinuous E J K) P)))
      (Functor.map (F.sheafPushforwardContinuous E J K)
          (NatTrans.app
            {
              app := fun (P : Sheaf.{v₂, v₃, u₂, u₃} K E) ↦
                { hom := NatTrans.app (unit (Adjunction.whiskerLeft E adj.op)) (P.obj (P := Presheaf.IsSheaf K)) },
              naturality := ⋯ }
            P) :
        Functor.obj (F.sheafPushforwardContinuous E J K) P ⟶
          Functor.obj (F.sheafPushforwardContinuous E J K)
            (Functor.obj (G.sheafPushforwardContinuous E K J) (Functor.obj (F.sheafPushforwardContinuous E J K) P)))
      (NatTrans.app
          {
            app := fun (P : Sheaf.{v₁, v₃, u₁, u₃} J E) ↦
              { hom := NatTrans.app (counit (Adjunction.whiskerLeft E adj.op)) (P.obj (P := Presheaf.IsSheaf J)) },
            naturality := ⋯ }
          (Functor.obj (F.sheafPushforwardContinuous E J K) P) :
        Functor.obj (G.sheafPushforwardContinuous E K J ⋙ F.sheafPushforwardContinuous E J K)
            (Functor.obj (F.sheafPushforwardContinuous E J K) P) ⟶
          Functor.obj (𝟭 (Sheaf.{v₁, v₃, u₁, u₃} J E)) (Functor.obj (F.sheafPushforwardContinuous E J K) P)) =
    𝟙 (Functor.obj (F.sheafPushforwardContinuous E J K) P)

Tactic:
ext : 1