-- Mathlib/CategoryTheory/SmallObject/Construction.lean
/-- Variant of `ιFunctorObj_extension` where the diagram involving `functorObj f πX`
is replaced by an isomorphic diagram. -/
lemma ιFunctorObj_extension' {X' S' Z' : C} (πX' : X' ⟶ S') (ι' : X' ⟶ Z') (πZ' : Z' ⟶ S')
    (fac' : ι' ≫ πZ' = πX') (eX : X' ≅ X) (eS : S' ≅ S) (eZ : Z' ≅ functorObj f πX)
    (commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX)
    (commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX)
    {i : I} (t : A i ⟶ X') (b : B i ⟶ S') (fac : t ≫ πX' = f i ≫ b) :
    ∃ (l : B i ⟶ Z'), f i ≫ l = t ≫ ι' ∧ l ≫ πZ' = b

Proof state:
case intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ ∃ l, f i ≫ l = t ≫ ι' ∧ l ≫ πZ' = b

Tactic:
refine ⟨l ≫ eZ.inv, ?_, ?_⟩

Result:
case intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ f i ≫ l ≫ eZ.inv = t ≫ ι'

case intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ (l ≫ eZ.inv) ≫ πZ' = b

Proof state:
case intro.intro.refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ (l ≫ eZ.inv) ≫ πZ' = b

Tactic:
rw [← cancel_mono eS.hom, assoc, assoc, commπ, eZ.inv_hom_id_assoc, hl₂]

Result:
no goals

Proof state:
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
⊢ ∃ l, f i ≫ l = t ≫ ι' ∧ l ≫ πZ' = b

Tactic:
obtain ⟨l, hl₁, hl₂⟩ :=
  ιFunctorObj_extension f (πX := πX) (i := i) (t ≫ eX.hom) (b ≫ eS.hom) ⟨by
    rw [assoc, ← ιFunctorObj_πFunctorObj f πX, ← reassoc_of% commι, ← commπ,
      reassoc_of% fac', reassoc_of% fac]⟩

Result:
case intro.intro
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ ∃ l, f i ≫ l = t ≫ ι' ∧ l ≫ πZ' = b

Proof state:
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
⊢ (t ≫ eX.hom) ≫ πX = f i ≫ b ≫ eS.hom

Tactic:
rw [assoc, ← ιFunctorObj_πFunctorObj f πX, ← reassoc_of% commι, ← commπ,
  reassoc_of% fac', reassoc_of% fac]

Result:
no goals

Proof state:
case intro.intro.refine_1
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ f i ≫ l ≫ eZ.inv = t ≫ ι'

Tactic:
rw [reassoc_of% hl₁, ← reassoc_of% commι, eZ.hom_inv_id, comp_id]

Result:
no goals