awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
a : J := IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
a' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
f :
  Total.mk (I := I) P (k.fst (β := fun (j : J) ↦ I j)) (k.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) a'
hf : f.base (I := I) = IsFiltered.leftToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
b' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
g :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) b'
hg : g.base (I := I) = IsFiltered.rightToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
⊢ k ⟶ Sigma.mk (β := fun (j : J) ↦ I j) a (IsFiltered.max a' b')

Tactic:
exact f ≫ { base := 𝟙 _, hom := (P _).diag.map (IsFiltered.leftToMax _ _) }
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
a : J := IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
a' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
f :
  Total.mk (I := I) P (k.fst (β := fun (j : J) ↦ I j)) (k.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) a'
hf : f.base (I := I) = IsFiltered.leftToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
b' : I (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j)))
g :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.max (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))) b'
hg : g.base (I := I) = IsFiltered.rightToMax (k.fst (β := fun (j : J) ↦ I j)) (l.fst (β := fun (j : J) ↦ I j))
⊢ l ⟶ Sigma.mk (β := fun (j : J) ↦ I j) a (IsFiltered.max a' b')

Tactic:
exact g ≫ { base := 𝟙 _, hom := (P _).diag.map (IsFiltered.rightToMax _ _) }
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
obtain :
  ∀ (a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
    (u :
      Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
        Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a')
    (hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))),
    ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h
⊢ ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h

Tactic:
obtain ⟨a', u, hu⟩ := Total.exists_hom_of_hom (P := P) l.2 (IsFiltered.coeqHom f.base g.base)

Premises used: CategoryTheory.Limits.ColimitPresentation.Total.exists_hom_of_hom, CategoryTheory.IsFiltered.coeqHom, CategoryTheory.Limits.ColimitPresentation.Total.Hom.base, Sigma.snd
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
simp :
  Eq (α :=
    Functor.obj (diag (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ⟶
      D.obj (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (f.base (I := I)) ≫ D.map (u.base (I := I)))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (g.base (I := I)) ≫ D.map (u.base (I := I)))
⊢ (f.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j)) =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j))

Tactic:
simp only [Category.assoc, Functor.const_obj_obj, ← u.w, ← f.w_assoc, ← g.w_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.ColimitPresentation.Total.Hom.w, CategoryTheory.Limits.ColimitPresentation.Total.Hom.w_assoc, CategoryTheory.Functor.const_obj_obj
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
rw :
  Eq (α :=
    Functor.obj (diag (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ⟶
      D.obj (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (g.base (I := I) ≫ IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (g.base (I := I)) ≫ D.map (IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))))
⊢ Eq (α :=
    Functor.obj (diag (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ⟶
      D.obj (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (f.base (I := I)) ≫ D.map (u.base (I := I)))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (g.base (I := I)) ≫ D.map (u.base (I := I)))

Tactic:
rw [← Functor.map_comp, hu, IsFiltered.coeq_condition f.base g.base]

Premises used: CategoryTheory.Limits.ColimitPresentation.Total.Hom.base, CategoryTheory.IsFiltered.coeq_condition, CategoryTheory.Functor.map_comp
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
⊢ Eq (α :=
    Functor.obj (diag (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ⟶
      D.obj (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (g.base (I := I) ≫ IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))))
    (NatTrans.app (ι (P (k.fst (β := fun (j : J) ↦ I j)))) (k.snd (β := fun (j : J) ↦ I j)) ≫
      D.map (g.base (I := I)) ≫ D.map (IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
⊢ (f.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j)) =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j))

Tactic:
have : (f.hom ≫ u.hom) ≫ (P _).ι.app _ = (g.hom ≫ u.hom) ≫ (P _).ι.app _ := by
      simp only [Category.assoc, Functor.const_obj_obj, ← u.w, ← f.w_assoc, ← g.w_assoc]
      rw [← Functor.map_comp, hu, IsFiltered.coeq_condition f.base g.base]
      simp
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
this :
  (f.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j)) =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j))
obtain :
  ∀
    (j :
      I ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦ I j)))
    (p q :
      (Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j) ⟶ j)
    (hpq :
      (f.hom (I := I) ≫ u.hom (I := I)) ≫
          Functor.map
            (diag
              (P
                ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
                  I j))))
            p =
        (g.hom (I := I) ≫ u.hom (I := I)) ≫
          Functor.map
            (diag
              (P
                ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
                  I j))))
            q),
    ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h
⊢ ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h

Tactic:
obtain ⟨j, p, q, hpq⟩ := IsFinitelyPresentable.exists_eq_of_isColimit (P _).isColimit _ _ this

Premises used: CategoryTheory.Limits.ColimitPresentation.isColimit, CategoryTheory.IsFinitelyPresentable.exists_eq_of_isColimit
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
this :
  (f.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j)) =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j))
j : I ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦ I j))
p q : a' ⟶ j
hpq :
  (f.hom (I := I) ≫ u.hom (I := I)) ≫
      Functor.map
        (diag
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        p =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      Functor.map
        (diag
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        q
⊢ ∃ (Z : Total (I := I) P) (h : l ⟶ Z), f ≫ h = g ≫ h

Tactic:
⟨⟨a, IsFiltered.coeq p q⟩,
      u ≫ { base := 𝟙 _, hom := (P _).diag.map (p ≫ IsFiltered.coeqHom p q) }, ?_⟩

Premises used: CategoryTheory.IsFiltered.coeqHom, CategoryTheory.Limits.ColimitPresentation.Total.Hom.base, CategoryTheory.CategoryStruct.id, Sigma.fst, CategoryTheory.Functor.map, CategoryTheory.Limits.ColimitPresentation.diag, CategoryTheory.Limits.ColimitPresentation.Total.Hom.mk, Exists.intro, CategoryTheory.CategoryStruct.comp, CategoryTheory.IsFiltered.coeq, CategoryTheory.Limits.ColimitPresentation.Total.mk, Sigma.mk
-- Module: Mathlib.CategoryTheory.Presentable.ColimitPresentation
Goal:
C : Type u
inst : Category.{v, u} C
J_1 : Type u_1
I_1 : J_1 → Type u_2
inst_1 : Category.{v_1, u_1} J_1
inst_2 : (j : J_1) → Category.{?u.30679, u_2} (I_1 j)
D_1 : Functor.{v_1, v, u_1, u} J_1 C
P_1 : (j : J_1) → ColimitPresentation.{?u.30679, u_2, v, u} (I_1 j) (D_1.obj j)
J : Type w
I : J → Type w
inst_3 : SmallCategory J
inst_4 : (j : J) → SmallCategory (I j)
D : Functor.{w, v, w, u} J C
P : (j : J) → ColimitPresentation.{w, w, v, u} (I j) (D.obj j)
inst_5 : IsFiltered J
inst_6 : ∀ (j : J), IsFiltered.{w, w} (I j)
inst_7 : ∀ (j : J) (i : I j), IsFinitelyPresentable.{w, v, u} ((P j).diag.obj i)
k l : Total (I := I) P
f g : k ⟶ l
a : J := IsFiltered.coeq (f.base (I := I)) (g.base (I := I))
a' : I (IsFiltered.coeq (f.base (I := I)) (g.base (I := I)))
u :
  Total.mk (I := I) P (l.fst (β := fun (j : J) ↦ I j)) (l.snd (β := fun (j : J) ↦ I j)) ⟶
    Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a'
hu : u.base (I := I) = IsFiltered.coeqHom (f.base (I := I)) (g.base (I := I))
this :
  (f.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j)) =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      NatTrans.app
        (ι
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').snd (β := fun (j : J) ↦ I j))
j : I ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦ I j))
p q : a' ⟶ j
hpq :
  (f.hom (I := I) ≫ u.hom (I := I)) ≫
      Functor.map
        (diag
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        p =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      Functor.map
        (diag
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        q
⊢ f.hom (I := I) ≫
      u.hom (I := I) ≫
        Functor.map
          (diag
            (P
              ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
                I j))))
          p =
    (g.hom (I := I) ≫ u.hom (I := I)) ≫
      Functor.map
        (diag
          (P
            ((Total.mk (I := I) P (IsFiltered.coeq (f.base (I := I)) (g.base (I := I))) a').fst (β := fun (j : J) ↦
              I j))))
        q

Tactic:
rw [Category.assoc] at hpq

Premises used: CategoryTheory.Category.assoc