-- Mathlib/CategoryTheory/SmallObject/Construction.lean
@[reassoc (attr := simp)]
lemma functorMap_π : functorMap f τ ≫ πFunctorObj f πY = πFunctorObj f πX ≫ τ.right

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S T X Y : C
πX : X ⟶ S
πY : Y ⟶ T
τ : Arrow.mk πX ⟶ Arrow.mk πY
inst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst✝¹ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
inst✝ : HasPushout (functorObjTop f πY) (functorObjLeft f πY)
⊢ functorMap f τ ≫ πFunctorObj f πY = πFunctorObj f πX ≫ τ.right

Tactic:
ext ⟨i, t, b, w⟩

Result:
case h₀
C : Type u
inst✝⁴ : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S T X Y : C
πX : X ⟶ S
πY : Y ⟶ T
τ : Arrow.mk πX ⟶ Arrow.mk πY
inst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst✝¹ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
inst✝ : HasPushout (functorObjTop f πY) (functorObjLeft f πY)
⊢ pushout.inl (functorObjTop f πX) (functorObjLeft f πX) ≫ functorMap f τ ≫ πFunctorObj f πY =
    pushout.inl (functorObjTop f πX) (functorObjLeft f πX) ≫ πFunctorObj f πX ≫ τ.right

case h₁.h.mk
C : Type u
inst✝⁴ : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S T X Y : C
πX : X ⟶ S
πY : Y ⟶ T
τ : Arrow.mk πX ⟶ Arrow.mk πY
inst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst✝¹ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
inst✝ : HasPushout (functorObjTop f πY) (functorObjLeft f πY)
i : I
t : A i ⟶ X
b : B i ⟶ S
w : t ≫ πX = f i ≫ b
⊢ Sigma.ι (functorObjTgtFamily f πX) { i := i, t := t, b := b, w := w } ≫
      pushout.inr (functorObjTop f πX) (functorObjLeft f πX) ≫ functorMap f τ ≫ πFunctorObj f πY =
    Sigma.ι (functorObjTgtFamily f πX) { i := i, t := t, b := b, w := w } ≫
      pushout.inr (functorObjTop f πX) (functorObjLeft f πX) ≫ πFunctorObj f πX ≫ τ.right

Proof state:
case h₀
C : Type u
inst✝⁴ : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S T X Y : C
πX : X ⟶ S
πY : Y ⟶ T
τ : Arrow.mk πX ⟶ Arrow.mk πY
inst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst✝¹ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
inst✝ : HasPushout (functorObjTop f πY) (functorObjLeft f πY)
⊢ pushout.inl (functorObjTop f πX) (functorObjLeft f πX) ≫ functorMap f τ ≫ πFunctorObj f πY =
    pushout.inl (functorObjTop f πX) (functorObjLeft f πX) ≫ πFunctorObj f πX ≫ τ.right

Tactic:
simp [functorMap]

Result:
no goals

Proof state:
case h₁.h.mk
C : Type u
inst✝⁴ : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S T X Y : C
πX : X ⟶ S
πY : Y ⟶ T
τ : Arrow.mk πX ⟶ Arrow.mk πY
inst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst✝¹ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
inst✝ : HasPushout (functorObjTop f πY) (functorObjLeft f πY)
i : I
t : A i ⟶ X
b : B i ⟶ S
w : t ≫ πX = f i ≫ b
⊢ Sigma.ι (functorObjTgtFamily f πX) { i := i, t := t, b := b, w := w } ≫
      pushout.inr (functorObjTop f πX) (functorObjLeft f πX) ≫ functorMap f τ ≫ πFunctorObj f πY =
    Sigma.ι (functorObjTgtFamily f πX) { i := i, t := t, b := b, w := w } ≫
      pushout.inr (functorObjTop f πX) (functorObjLeft f πX) ≫ πFunctorObj f πX ≫ τ.right

Tactic:
simp [functorMap, ι_functorMapTgt_assoc f τ i t b w _ rfl]

Result:
no goals