l3lab/ntp-mathlib huggingface dataset:
/-- The functor sending each quiver to its path category. -/
@[simps]
def free : Quiv.{v, u} ⥤ Cat.{max u v, u}
    where
  obj V := Cat.of (Paths V)
  map F :=
    { obj := fun X => F.obj X
      map := fun f => F.mapPath f
      map_comp := fun f g => F.mapPath_comp f g }
  map_id V := by
    change (show Paths V ⥤ _ from _) = _
    ext; 

Proof state:
case h_obj.h
V : Quiv
x✝ : Paths ↑V
⊢ (let_fun this :=
          { obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
            (𝟙 V);
        this).obj
      x✝ =
    (𝟙
          ({ obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
            V)).obj
      x✝
case h
V : Quiv
a✝ b✝ : ↑V
e✝ : a✝ ⟶ b✝
⊢ (let_fun this :=
          { obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
            (𝟙 V);
        this).map
      (Quiver.Hom.toPath e✝) =
    eqToHom
        (_ :
          (let_fun this :=
                  { obj := fun V => of (Paths ↑V),
                        map := fun {X Y} F =>
                          Functor.mk
                            { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
                    (𝟙 V);
                this).obj
              a✝ =
            (𝟙
                  ({ obj := fun V => of (Paths ↑V),
                        map := fun {X Y} F =>
                          Functor.mk
                            { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
                    V)).obj
              a✝) ≫
      (𝟙
              ({ obj := fun V => of (Paths ↑V),
                    map := fun {X Y} F =>
                      Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
                V)).map
          (Quiver.Hom.toPath e✝) ≫
        eqToHom
          (_ :
            (𝟙
                    ({ obj := fun V => of (Paths ↑V),
                          map := fun {X Y} F =>
                            Functor.mk
                              { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
                      V)).obj
                b✝ =
              (let_fun this :=
                    { obj := fun V => of (Paths ↑V),
                          map := fun {X Y} F =>
                            Functor.mk
                              { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
                      (𝟙 V);
                  this).obj
                b✝)

Next tactic:
swap
/-- The functor sending each quiver to its path category. -/
@[simps]
def free : Quiv.{v, u} ⥤ Cat.{max u v, u}
    where
  obj V := Cat.of (Paths V)
  map F :=
    { obj := fun X => F.obj X
      map := fun f => F.mapPath f
      map_comp := fun f g => F.mapPath_comp f g }
  map_id V := by
    change (show Paths V ⥤ _ from _) = _
    ext; swap
    

Proof state:
case h
V : Quiv
a✝ b✝ : ↑V
e✝ : a✝ ⟶ b✝
⊢ (let_fun this :=
          { obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
            (𝟙 V);
        this).map
      (Quiver.Hom.toPath e✝) =
    eqToHom
        (_ :
          (let_fun this :=
                  { obj := fun V => of (Paths ↑V),
                        map := fun {X Y} F =>
                          Functor.mk
                            { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
                    (𝟙 V);
                this).obj
              a✝ =
            (𝟙
                  ({ obj := fun V => of (Paths ↑V),
                        map := fun {X Y} F =>
                          Functor.mk
                            { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
                    V)).obj
              a✝) ≫
      (𝟙
              ({ obj := fun V => of (Paths ↑V),
                    map := fun {X Y} F =>
                      Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
                V)).map
          (Quiver.Hom.toPath e✝) ≫
        eqToHom
          (_ :
            (𝟙
                    ({ obj := fun V => of (Paths ↑V),
                          map := fun {X Y} F =>
                            Functor.mk
                              { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
                      V)).obj
                b✝ =
              (let_fun this :=
                    { obj := fun V => of (Paths ↑V),
                          map := fun {X Y} F =>
                            Functor.mk
                              { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
                      (𝟙 V);
                  this).obj
                b✝)
case h_obj.h
V : Quiv
x✝ : Paths ↑V
⊢ (let_fun this :=
          { obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
            (𝟙 V);
        this).obj
      x✝ =
    (𝟙
          ({ obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
            V)).obj
      x✝

Next tactic:
apply eq_conj_eqToHom
/-- The functor sending each quiver to its path category. -/
@[simps]
def free : Quiv.{v, u} ⥤ Cat.{max u v, u}
    where
  obj V := Cat.of (Paths V)
  map F :=
    { obj := fun X => F.obj X
      map := fun f => F.mapPath f
      map_comp := fun f g => F.mapPath_comp f g }
  map_id V := by
    change (show Paths V ⥤ _ from _) = _
    ext; swap
    apply eq_conj_eqToHom
    

Proof state:
case h_obj.h
V : Quiv
x✝ : Paths ↑V
⊢ (let_fun this :=
          { obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
            (𝟙 V);
        this).obj
      x✝ =
    (𝟙
          ({ obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.obj
            V)).obj
      x✝

Next tactic:
rfl
/-- The functor sending each quiver to its path category. -/
@[simps]
def free : Quiv.{v, u} ⥤ Cat.{max u v, u}
    where
  obj V := Cat.of (Paths V)
  map F :=
    { obj := fun X => F.obj X
      map := fun f => F.mapPath f
      map_comp := fun f g => F.mapPath_comp f g }
  map_id V := by
    change (show Paths V ⥤ _ from _) = _
    ext; swap
    apply eq_conj_eqToHom
    rfl
  map_comp {U _ _} F G := by
    

Proof state:
U x✝¹ x✝ : Quiv
F : U ⟶ x✝¹
G : x✝¹ ⟶ x✝
⊢ { obj := fun V => of (Paths ↑V),
          map := fun {X Y} F =>
            Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
      (F ≫ G) =
    { obj := fun V => of (Paths ↑V),
            map := fun {X Y} F =>
              Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
        F ≫
      { obj := fun V => of (Paths ↑V),
            map := fun {X Y} F =>
              Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
        G

Next tactic:
change (show Paths U ⥤ _ from _) = _
/-- The functor sending each quiver to its path category. -/
@[simps]
def free : Quiv.{v, u} ⥤ Cat.{max u v, u}
    where
  obj V := Cat.of (Paths V)
  map F :=
    { obj := fun X => F.obj X
      map := fun f => F.mapPath f
      map_comp := fun f g => F.mapPath_comp f g }
  map_id V := by
    change (show Paths V ⥤ _ from _) = _
    ext; swap
    apply eq_conj_eqToHom
    rfl
  map_comp {U _ _} F G := by
    change (show Paths U ⥤ _ from _) = _
    

Proof state:
U x✝¹ x✝ : Quiv
F : U ⟶ x✝¹
G : x✝¹ ⟶ x✝
⊢ (let_fun this :=
      { obj := fun V => of (Paths ↑V),
            map := fun {X Y} F =>
              Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
        (F ≫ G);
    this) =
    { obj := fun V => of (Paths ↑V),
            map := fun {X Y} F =>
              Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
        F ≫
      { obj := fun V => of (Paths ↑V),
            map := fun {X Y} F =>
              Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
        G

Next tactic:
ext
/-- The functor sending each quiver to its path category. -/
@[simps]
def free : Quiv.{v, u} ⥤ Cat.{max u v, u}
    where
  obj V := Cat.of (Paths V)
  map F :=
    { obj := fun X => F.obj X
      map := fun f => F.mapPath f
      map_comp := fun f g => F.mapPath_comp f g }
  map_id V := by
    change (show Paths V ⥤ _ from _) = _
    ext; swap
    apply eq_conj_eqToHom
    rfl
  map_comp {U _ _} F G := by
    change (show Paths U ⥤ _ from _) = _
    ext; swap
    apply eq_conj_eqToHom
    

Proof state:
case h_obj.h
U x✝² x✝¹ : Quiv
F : U ⟶ x✝²
G : x✝² ⟶ x✝¹
x✝ : Paths ↑U
⊢ (let_fun this :=
          { obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
            (F ≫ G);
        this).obj
      x✝ =
    ({ obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
            F ≫
          { obj := fun V => of (Paths ↑V),
                map := fun {X Y} F =>
                  Functor.mk { obj := fun X_1 => F.obj X_1, map := fun {X_1 Y_1} f => Prefunctor.mapPath F f } }.map
            G).obj
      x✝

Next tactic:
rfl
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by 

Proof state:
V : Quiv
C : Cat
F : Cat.free.obj V ⟶ C
⊢ ∀ (a b : ↑V) (e : a ⟶ b),
    ((fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F)).map (Quiver.Hom.toPath e) =
      eqToHom (_ : ((fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F)).obj a = F.obj a) ≫
        F.map (Quiver.Hom.toPath e) ≫
          eqToHom (_ : F.obj b = ((fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F)).obj b)

Next tactic:
simp
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by simp)
          right_inv := by
            

Proof state:
V : Quiv
C : Cat
⊢ Function.RightInverse (fun F => lift F) fun F => Paths.of ⋙q F.toPrefunctor

Next tactic:
rintro ⟨obj, map⟩
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by simp)
          right_inv := by
            rintro ⟨obj, map⟩
            

Proof state:
case mk
V : Quiv
C : Cat
obj : ↑V → ↑(forget.obj C)
map : {X Y : ↑V} → (X ⟶ Y) → (obj X ⟶ obj Y)
⊢ (fun F => Paths.of ⋙q F.toPrefunctor) ((fun F => lift F) { obj := obj, map := map }) = { obj := obj, map := map }

Next tactic:
dsimp only [Prefunctor.comp]
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by simp)
          right_inv := by
            rintro ⟨obj, map⟩
            dsimp only [Prefunctor.comp]
            

Proof state:
case mk
V : Quiv
C : Cat
obj : ↑V → ↑(forget.obj C)
map : {X Y : ↑V} → (X ⟶ Y) → (obj X ⟶ obj Y)
⊢ { obj := fun X => (lift { obj := obj, map := map }).obj (Paths.of.obj X),
      map := fun {X Y} f => (lift { obj := obj, map := map }).map (Paths.of.map f) } =
    { obj := obj, map := map }

Next tactic:
congr
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by simp)
          right_inv := by
            rintro ⟨obj, map⟩
            dsimp only [Prefunctor.comp]
            congr
            

Proof state:
case mk.e_map
V : Quiv
C : Cat
obj : ↑V → ↑(forget.obj C)
map : {X Y : ↑V} → (X ⟶ Y) → (obj X ⟶ obj Y)
⊢ (fun {X Y} f => (lift { obj := obj, map := map }).map (Paths.of.map f)) = map

Next tactic:
funext X Y f
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by simp)
          right_inv := by
            rintro ⟨obj, map⟩
            dsimp only [Prefunctor.comp]
            congr
            funext X Y f
            

Proof state:
case mk.e_map.h.h.h
V : Quiv
C : Cat
obj : ↑V → ↑(forget.obj C)
map : {X Y : ↑V} → (X ⟶ Y) → (obj X ⟶ obj Y)
X Y : ↑V
f : X ⟶ Y
⊢ (lift { obj := obj, map := map }).map (Paths.of.map f) = map f

Next tactic:
exact Category.id_comp _
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by simp)
          right_inv := by
            rintro ⟨obj, map⟩
            dsimp only [Prefunctor.comp]
            congr
            funext X Y f
            exact Category.id_comp _ }
      homEquiv_naturality_left_symm := fun {V _ _} f g => by
        

Proof state:
V x✝¹ : Quiv
x✝ : Cat
f : V ⟶ x✝¹
g : x✝¹ ⟶ forget.obj x✝
⊢ ((fun V C =>
            { toFun := fun F => Paths.of ⋙q F.toPrefunctor, invFun := fun F => lift F,
              left_inv :=
                (_ : ∀ (F : Cat.free.obj V ⟶ C), (fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F) = F),
              right_inv :=
                (_ : ∀ (x : V ⟶ forget.obj C), (fun F => Paths.of ⋙q F.toPrefunctor) ((fun F => lift F) x) = x) })
          V x✝).symm
      (f ≫ g) =
    Cat.free.map f ≫
      ((fun V C =>
              { toFun := fun F => Paths.of ⋙q F.toPrefunctor, invFun := fun F => lift F,
                left_inv :=
                  (_ : ∀ (F : Cat.free.obj V ⟶ C), (fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F) = F),
                right_inv :=
                  (_ : ∀ (x : V ⟶ forget.obj C), (fun F => Paths.of ⋙q F.toPrefunctor) ((fun F => lift F) x) = x) })
            x✝¹ x✝).symm
        g

Next tactic:
change (show Paths V ⥤ _ from _) = _
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by simp)
          right_inv := by
            rintro ⟨obj, map⟩
            dsimp only [Prefunctor.comp]
            congr
            funext X Y f
            exact Category.id_comp _ }
      homEquiv_naturality_left_symm := fun {V _ _} f g => by
        change (show Paths V ⥤ _ from _) = _
        

Proof state:
V x✝¹ : Quiv
x✝ : Cat
f : V ⟶ x✝¹
g : x✝¹ ⟶ forget.obj x✝
⊢ (let_fun this :=
      ((fun V C =>
              { toFun := fun F => Paths.of ⋙q F.toPrefunctor, invFun := fun F => lift F,
                left_inv :=
                  (_ : ∀ (F : Cat.free.obj V ⟶ C), (fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F) = F),
                right_inv :=
                  (_ : ∀ (x : V ⟶ forget.obj C), (fun F => Paths.of ⋙q F.toPrefunctor) ((fun F => lift F) x) = x) })
            V x✝).symm
        (f ≫ g);
    this) =
    Cat.free.map f ≫
      ((fun V C =>
              { toFun := fun F => Paths.of ⋙q F.toPrefunctor, invFun := fun F => lift F,
                left_inv :=
                  (_ : ∀ (F : Cat.free.obj V ⟶ C), (fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F) = F),
                right_inv :=
                  (_ : ∀ (x : V ⟶ forget.obj C), (fun F => Paths.of ⋙q F.toPrefunctor) ((fun F => lift F) x) = x) })
            x✝¹ x✝).symm
        g

Next tactic:
ext
/--
The adjunction between forming the free category on a quiver, and forgetting a category to a quiver.
-/
def adj : Cat.free ⊣ Quiv.forget :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun V C =>
        { toFun := fun F => Paths.of.comp F.toPrefunctor
          invFun := fun F => @lift V _ C _ F
          left_inv := fun F => Paths.ext_functor rfl (by simp)
          right_inv := by
            rintro ⟨obj, map⟩
            dsimp only [Prefunctor.comp]
            congr
            funext X Y f
            exact Category.id_comp _ }
      homEquiv_naturality_left_symm := fun {V _ _} f g => by
        change (show Paths V ⥤ _ from _) = _
        ext; swap
        apply eq_conj_eqToHom
        

Proof state:
case h_obj.h
V x✝² : Quiv
x✝¹ : Cat
f : V ⟶ x✝²
g : x✝² ⟶ forget.obj x✝¹
x✝ : Paths ↑V
⊢ (let_fun this :=
          ((fun V C =>
                  { toFun := fun F => Paths.of ⋙q F.toPrefunctor, invFun := fun F => lift F,
                    left_inv :=
                      (_ : ∀ (F : Cat.free.obj V ⟶ C), (fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F) = F),
                    right_inv :=
                      (_ : ∀ (x : V ⟶ forget.obj C), (fun F => Paths.of ⋙q F.toPrefunctor) ((fun F => lift F) x) = x) })
                V x✝¹).symm
            (f ≫ g);
        this).obj
      x✝ =
    (Cat.free.map f ≫
          ((fun V C =>
                  { toFun := fun F => Paths.of ⋙q F.toPrefunctor, invFun := fun F => lift F,
                    left_inv :=
                      (_ : ∀ (F : Cat.free.obj V ⟶ C), (fun F => lift F) ((fun F => Paths.of ⋙q F.toPrefunctor) F) = F),
                    right_inv :=
                      (_ : ∀ (x : V ⟶ forget.obj C), (fun F => Paths.of ⋙q F.toPrefunctor) ((fun F => lift F) x) = x) })
                x✝² x✝¹).symm
            g).obj
      x✝

Next tactic:
rfl
instance (F : LaxMonoidalFunctor V W) : EnrichedCategory W (TransportEnrichment F C) where
  Hom := fun X Y : C => F.obj (X ⟶[V] Y)
  id := fun X : C => F.ε ≫ F.map (eId V X)
  comp := fun X Y Z : C => F.μ _ _ ≫ F.map (eComp V X Y Z)
  id_comp X Y := by
    

Proof state:
V : Type v
inst✝⁴ : Category.{w, v} V
inst✝³ : MonoidalCategory V
C : Type u₁
inst✝² : EnrichedCategory V C
W : Type v
inst✝¹ : Category.{w, v} W
inst✝ : MonoidalCategory W
F : LaxMonoidalFunctor V W
X Y : TransportEnrichment F C
⊢ (λ_ ((fun X Y => F.obj (EnrichedCategory.Hom X Y)) X Y)).inv ≫
      ((fun X => F.ε ≫ F.map (eId V X)) X ⊗ 𝟙 ((fun X Y => F.obj (EnrichedCategory.Hom X Y)) X Y)) ≫
        (fun X Y Z =>
            LaxMonoidalFunctor.μ F (EnrichedCategory.Hom X Y) (EnrichedCategory.Hom Y Z) ≫ F.map (eComp V X Y Z))
          X X Y =
    𝟙 ((fun X Y => F.obj (EnrichedCategory.Hom X Y)) X Y)

Next tactic:
rw [comp_tensor_id, Category.assoc, ← F.toFunctor.map_id, F.μ_natural_assoc,
      F.toFunctor.map_id, F.left_unitality_inv_assoc, ← F.toFunctor.map_comp, ←
      F.toFunctor.map_comp, e_id_comp, F.toFunctor.map_id]
instance (F : LaxMonoidalFunctor V W) : EnrichedCategory W (TransportEnrichment F C) where
  Hom := fun X Y : C => F.obj (X ⟶[V] Y)
  id := fun X : C => F.ε ≫ F.map (eId V X)
  comp := fun X Y Z : C => F.μ _ _ ≫ F.map (eComp V X Y Z)
  id_comp X Y := by
    rw [comp_tensor_id, Category.assoc, ← F.toFunctor.map_id, F.μ_natural_assoc,
      F.toFunctor.map_id, F.left_unitality_inv_assoc, ← F.toFunctor.map_comp, ←
      F.toFunctor.map_comp, e_id_comp, F.toFunctor.map_id]
  comp_id X Y := by
    

Proof state:
V : Type v
inst✝⁴ : Category.{w, v} V
inst✝³ : MonoidalCategory V
C : Type u₁
inst✝² : EnrichedCategory V C
W : Type v
inst✝¹ : Category.{w, v} W
inst✝ : MonoidalCategory W
F : LaxMonoidalFunctor V W
X Y : TransportEnrichment F C
⊢ (ρ_ ((fun X Y => F.obj (EnrichedCategory.Hom X Y)) X Y)).inv ≫
      (𝟙 ((fun X Y => F.obj (EnrichedCategory.Hom X Y)) X Y) ⊗ (fun X => F.ε ≫ F.map (eId V X)) Y) ≫
        (fun X Y Z =>
            LaxMonoidalFunctor.μ F (EnrichedCategory.Hom X Y) (EnrichedCategory.Hom Y Z) ≫ F.map (eComp V X Y Z))
          X Y Y =
    𝟙 ((fun X Y => F.obj (EnrichedCategory.Hom X Y)) X Y)

Next tactic:
rw [id_tensor_comp, Category.assoc, ← F.toFunctor.map_id, F.μ_natural_assoc,
      F.toFunctor.map_id, F.right_unitality_inv_assoc, ← F.toFunctor.map_comp, ←
      F.toFunctor.map_comp, e_comp_id, F.toFunctor.map_id]