V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
⊢ Φ =
    { obj := τ,
      map :=
        @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
          id
            (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
              (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
⊢ Φ = lift σ φ τ h
dsimp only [of, lift]
case h_obj
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
⊢ ∀ (X : Push σ),
    Φ.obj X =
      { obj := τ,
            map :=
              @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                id
                  (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                    (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
        X

case h_map
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
⊢ ∀ (X Y : Push σ) (f : X ⟶ Y),
    Φ.map f =
      Eq.recOn
        (_ :
          { obj := τ,
                  map :=
                    @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                      id
                        (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                          (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
              Y =
            Φ.obj Y)
        (Eq.recOn
          (_ :
            { obj := τ,
                    map :=
                      @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                        id
                          (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                            (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
                X =
              Φ.obj X)
          ({ obj := τ,
                map :=
                  @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                    id
                      (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                        (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.map
            f))
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
⊢ Φ =
    { obj := τ,
      map :=
        @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
          id
            (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
              (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }
fapply Prefunctor.ext
case h_obj
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
X : Push σ
⊢ Φ.obj X =
    { obj := τ,
          map :=
            @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
              id
                (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                  (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
      X
case h_obj
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
⊢ ∀ (X : Push σ),
    Φ.obj X =
      { obj := τ,
            map :=
              @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                id
                  (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                    (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
        X
intro X
case h_obj
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
X : Push σ
⊢ Φ.obj X = τ X
case h_obj
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
X : Push σ
⊢ Φ.obj X =
    { obj := τ,
          map :=
            @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
              id
                (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                  (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
      X
simp only
no goals
case h_obj
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
X : Push σ
⊢ Φ.obj X = τ X
rw [Φ₀]
case h_map.arrow
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
X✝ Y✝ : V
f✝ : X✝ ⟶ Y✝
⊢ Φ.map (PushQuiver.arrow f✝) =
    Eq.recOn
      (_ :
        { obj := τ,
                map :=
                  @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                    id
                      (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                        (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
            (σ Y✝) =
          Φ.obj (σ Y✝))
      (Eq.recOn
        (_ :
          { obj := τ,
                  map :=
                    @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                      id
                        (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                          (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
              (σ X✝) =
            Φ.obj (σ X✝))
        ({ obj := τ,
              map :=
                @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                  id
                    (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                      (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.map
          (PushQuiver.arrow f✝)))
case h_map
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
⊢ ∀ (X Y : Push σ) (f : X ⟶ Y),
    Φ.map f =
      Eq.recOn
        (_ :
          { obj := τ,
                  map :=
                    @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                      id
                        (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                          (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
              Y =
            Φ.obj Y)
        (Eq.recOn
          (_ :
            { obj := τ,
                    map :=
                      @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                        id
                          (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                            (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
                X =
              Φ.obj X)
          ({ obj := τ,
                map :=
                  @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                    id
                      (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                        (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.map
            f))
rintro _ _ ⟨⟩
case h_map.arrow
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
Φ : Push σ ⥤q W'
X✝ Y✝ : V
f✝ : X✝ ⟶ Y✝
h : ∀ (x : V), (of σ ⋙q Φ).obj x = Φ.obj (σ x)
⊢ Φ.map (PushQuiver.arrow f✝) =
    Eq.recOn
      (_ :
        { obj := Φ.obj,
                map :=
                  @PushQuiver.rec V inst✝¹ W σ (fun X Y x => Φ.obj X ⟶ Φ.obj Y) fun X Y f =>
                    id
                      (Eq.mpr (_ : (Φ.obj (σ X) ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)))
                        (Eq.mpr (_ : ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ (of σ ⋙q Φ).obj Y))
                          ((of σ ⋙q Φ).map f))) }.obj
            (σ Y✝) =
          Φ.obj (σ Y✝))
      (Eq.recOn
        (_ :
          { obj := Φ.obj,
                  map :=
                    @PushQuiver.rec V inst✝¹ W σ (fun X Y x => Φ.obj X ⟶ Φ.obj Y) fun X Y f =>
                      id
                        (Eq.mpr (_ : (Φ.obj (σ X) ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)))
                          (Eq.mpr (_ : ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ (of σ ⋙q Φ).obj Y))
                            ((of σ ⋙q Φ).map f))) }.obj
              (σ X✝) =
            Φ.obj (σ X✝))
        ({ obj := Φ.obj,
              map :=
                @PushQuiver.rec V inst✝¹ W σ (fun X Y x => Φ.obj X ⟶ Φ.obj Y) fun X Y f =>
                  id
                    (Eq.mpr (_ : (Φ.obj (σ X) ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)))
                      (Eq.mpr (_ : ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ (of σ ⋙q Φ).obj Y))
                        ((of σ ⋙q Φ).map f))) }.map
          (PushQuiver.arrow f✝)))
case h_map.arrow
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
Φ : Push σ ⥤q W'
Φ₀ : Φ.obj = τ
Φcomp : of σ ⋙q Φ = φ
X✝ Y✝ : V
f✝ : X✝ ⟶ Y✝
⊢ Φ.map (PushQuiver.arrow f✝) =
    Eq.recOn
      (_ :
        { obj := τ,
                map :=
                  @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                    id
                      (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                        (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
            (σ Y✝) =
          Φ.obj (σ Y✝))
      (Eq.recOn
        (_ :
          { obj := τ,
                  map :=
                    @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                      id
                        (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                          (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.obj
              (σ X✝) =
            Φ.obj (σ X✝))
        ({ obj := τ,
              map :=
                @PushQuiver.rec V inst✝¹ W σ (fun X Y x => τ X ⟶ τ Y) fun X Y f =>
                  id
                    (Eq.mpr (_ : (τ (σ X) ⟶ τ (σ Y)) = (φ.obj X ⟶ τ (σ Y)))
                      (Eq.mpr (_ : (φ.obj X ⟶ τ (σ Y)) = (φ.obj X ⟶ φ.obj Y)) (φ.map f))) }.map
          (PushQuiver.arrow f✝)))
subst_vars
case h_map.arrow
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
Φ : Push σ ⥤q W'
X✝ Y✝ : V
f✝ : X✝ ⟶ Y✝
h : ∀ (x : V), (of σ ⋙q Φ).obj x = Φ.obj (σ x)
⊢ Φ.map (PushQuiver.arrow f✝) =
    id
      (Eq.mpr (_ : (Φ.obj (σ X✝) ⟶ Φ.obj (σ Y✝)) = ((of σ ⋙q Φ).obj X✝ ⟶ Φ.obj (σ Y✝)))
        (Eq.mpr (_ : ((of σ ⋙q Φ).obj X✝ ⟶ Φ.obj (σ Y✝)) = ((of σ ⋙q Φ).obj X✝ ⟶ (of σ ⋙q Φ).obj Y✝))
          (Φ.map ((of σ).map f✝))))
case h_map.arrow
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
Φ : Push σ ⥤q W'
X✝ Y✝ : V
f✝ : X✝ ⟶ Y✝
h : ∀ (x : V), (of σ ⋙q Φ).obj x = Φ.obj (σ x)
⊢ Φ.map (PushQuiver.arrow f✝) =
    Eq.recOn
      (_ :
        { obj := Φ.obj,
                map :=
                  @PushQuiver.rec V inst✝¹ W σ (fun X Y x => Φ.obj X ⟶ Φ.obj Y) fun X Y f =>
                    id
                      (Eq.mpr (_ : (Φ.obj (σ X) ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)))
                        (Eq.mpr (_ : ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ (of σ ⋙q Φ).obj Y))
                          ((of σ ⋙q Φ).map f))) }.obj
            (σ Y✝) =
          Φ.obj (σ Y✝))
      (Eq.recOn
        (_ :
          { obj := Φ.obj,
                  map :=
                    @PushQuiver.rec V inst✝¹ W σ (fun X Y x => Φ.obj X ⟶ Φ.obj Y) fun X Y f =>
                      id
                        (Eq.mpr (_ : (Φ.obj (σ X) ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)))
                          (Eq.mpr (_ : ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ (of σ ⋙q Φ).obj Y))
                            ((of σ ⋙q Φ).map f))) }.obj
              (σ X✝) =
            Φ.obj (σ X✝))
        ({ obj := Φ.obj,
              map :=
                @PushQuiver.rec V inst✝¹ W σ (fun X Y x => Φ.obj X ⟶ Φ.obj Y) fun X Y f =>
                  id
                    (Eq.mpr (_ : (Φ.obj (σ X) ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)))
                      (Eq.mpr (_ : ((of σ ⋙q Φ).obj X ⟶ Φ.obj (σ Y)) = ((of σ ⋙q Φ).obj X ⟶ (of σ ⋙q Φ).obj Y))
                        ((of σ ⋙q Φ).map f))) }.map
          (PushQuiver.arrow f✝)))
simp only [Prefunctor.comp_map, cast_eq]
no goals
case h_map.arrow
V : Type u_1
inst✝¹ : Quiver V
W : Type u_3
σ : V → W
W' : Type u_5
inst✝ : Quiver W'
Φ : Push σ ⥤q W'
X✝ Y✝ : V
f✝ : X✝ ⟶ Y✝
h : ∀ (x : V), (of σ ⋙q Φ).obj x = Φ.obj (σ x)
⊢ Φ.map (PushQuiver.arrow f✝) =
    id
      (Eq.mpr (_ : (Φ.obj (σ X✝) ⟶ Φ.obj (σ Y✝)) = ((of σ ⋙q Φ).obj X✝ ⟶ Φ.obj (σ Y✝)))
        (Eq.mpr (_ : ((of σ ⋙q Φ).obj X✝ ⟶ Φ.obj (σ Y✝)) = ((of σ ⋙q Φ).obj X✝ ⟶ (of σ ⋙q Φ).obj Y✝))
          (Φ.map ((of σ).map f✝))))
rfl