-- Mathlib/CategoryTheory/Limits/Constructions/WeaklyInitial.lean
theorem hasInitial_of_weakly_initial_and_hasWideEqualizers [HasWideEqualizers.{v} C] {T : C}
    (hT : ∀ X, Nonempty (T ⟶ X)) : HasInitial C

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this✝ : Nonempty endos
X : C
a : wideEqualizer id ⟶ X
E : C := equalizer a (i ≫ Classical.choice ⋯)
e : E ⟶ wideEqualizer id := equalizer.ι a (i ≫ Classical.choice ⋯)
h : T ⟶ E := Classical.choice ⋯
this : ((i ≫ h) ≫ e) ≫ i = i ≫ 𝟙 T
⊢ a = default

Tactic:
rw [Category.comp_id, cancel_mono_id i] at this

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this✝ : Nonempty endos
X : C
a : wideEqualizer id ⟶ X
E : C := equalizer a (i ≫ Classical.choice ⋯)
e : E ⟶ wideEqualizer id := equalizer.ι a (i ≫ Classical.choice ⋯)
h : T ⟶ E := Classical.choice ⋯
this : (i ≫ h) ≫ e = 𝟙 (wideEqualizer id)
⊢ a = default

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this : Nonempty endos
X : C
a : wideEqualizer id ⟶ X
E : C := equalizer a (i ≫ Classical.choice ⋯)
e : E ⟶ wideEqualizer id := equalizer.ι a (i ≫ Classical.choice ⋯)
h : T ⟶ E := Classical.choice ⋯
⊢ a = default

Tactic:
have : ((i ≫ h) ≫ e) ≫ i = i ≫ 𝟙 _ := by
  rw [Category.assoc, Category.assoc]
  apply wideEqualizer.condition (id : endos → endos) (h ≫ e ≫ i)

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this✝ : Nonempty endos
X : C
a : wideEqualizer id ⟶ X
E : C := equalizer a (i ≫ Classical.choice ⋯)
e : E ⟶ wideEqualizer id := equalizer.ι a (i ≫ Classical.choice ⋯)
h : T ⟶ E := Classical.choice ⋯
this : ((i ≫ h) ≫ e) ≫ i = i ≫ 𝟙 T
⊢ a = default

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
⊢ HasInitial C

Tactic:
let endos := T ⟶ T

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
⊢ HasInitial C

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this : Nonempty endos
X : C
a : wideEqualizer id ⟶ X
E : C := equalizer a (i ≫ Classical.choice ⋯)
e : E ⟶ wideEqualizer id := equalizer.ι a (i ≫ Classical.choice ⋯)
⊢ a = default

Tactic:
let h : T ⟶ E := Classical.choice (hT E)

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this : Nonempty endos
X : C
a : wideEqualizer id ⟶ X
E : C := equalizer a (i ≫ Classical.choice ⋯)
e : E ⟶ wideEqualizer id := equalizer.ι a (i ≫ Classical.choice ⋯)
h : T ⟶ E := Classical.choice ⋯
⊢ a = default

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this : Nonempty endos
X : C
⊢ Unique (wideEqualizer id ⟶ X)

Tactic:
refine ⟨⟨i ≫ Classical.choice (hT X)⟩, fun a => ?_⟩

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this : Nonempty endos
X : C
a : wideEqualizer id ⟶ X
⊢ a = default

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
⊢ HasInitial C

Tactic:
let i := wideEqualizer.ι (id : endos → endos)

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
⊢ HasInitial C

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this : Nonempty endos
⊢ (X : C) → Unique (wideEqualizer id ⟶ X)

Tactic:
intro X

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasWideEqualizers C
T : C
hT : ∀ (X : C), Nonempty (T ⟶ X)
endos : Type v := T ⟶ T
i : wideEqualizer id ⟶ T := wideEqualizer.ι id
this : Nonempty endos
X : C
⊢ Unique (wideEqualizer id ⟶ X)