-- Module: Mathlib.CategoryTheory.SmallObject.Construction
Goal:
C : Type u
inst : 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_1 : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst_2 : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C
inst_3 :
  HasPushout (X := ∐ functorObjSrcFamily f πX) (Z := ∐ functorObjTgtFamily f πX)
    (functorObjTop f πX : ∐ functorObjSrcFamily f πX ⟶ X)
    (functorObjLeft f πX : ∐ functorObjSrcFamily f πX ⟶ ∐ functorObjTgtFamily f πX)
inst_4 :
  HasPushout (X := ∐ functorObjSrcFamily f πY) (Z := ∐ functorObjTgtFamily f πY)
    (functorObjTop f πY : ∐ functorObjSrcFamily f πY ⟶ Y)
    (functorObjLeft f πY : ∐ functorObjSrcFamily f πY ⟶ ∐ functorObjTgtFamily f πY)
i : I
t : A i ⟶ X
b : B i ⟶ S
w : t ≫ πX = f i ≫ b
⊢ Eq (α := functorObjTgtFamily f πX { i := i, t := t, b := b, w := w : FunctorObjIndex f πX } ⟶ T)
    (CategoryStruct.comp (Y := ∐ functorObjTgtFamily f πX)
      (Sigma.ι (functorObjTgtFamily f πX) { i := i, t := t, b := b, w := w } :
        functorObjTgtFamily f πX { i := i, t := t, b := b, w := w : FunctorObjIndex f πX } ⟶ ∐ functorObjTgtFamily f πX)
      (CategoryStruct.comp (Y :=
          pushout (X := ∐ functorObjSrcFamily f πX) (Z := ∐ functorObjTgtFamily f πX)
            (functorObjTop f πX : ∐ functorObjSrcFamily f πX ⟶ X)
            (functorObjLeft f πX : ∐ functorObjSrcFamily f πX ⟶ ∐ functorObjTgtFamily f πX))
          (pushout.inr (functorObjTop f πX) (functorObjLeft f πX) :
            ∐ functorObjTgtFamily f πX ⟶
              pushout (X := ∐ functorObjSrcFamily f πX) (Z := ∐ functorObjTgtFamily f πX)
                (functorObjTop f πX : ∐ functorObjSrcFamily f πX ⟶ X)
                (functorObjLeft f πX : ∐ functorObjSrcFamily f πX ⟶ ∐ functorObjTgtFamily f πX))
          (CategoryStruct.comp (Y := functorObj f πY) (functorMap f τ) (πFunctorObj f πY) : functorObj f πX ⟶ T) :
        ∐ functorObjTgtFamily f πX ⟶ T))
    (CategoryStruct.comp (Y := ∐ functorObjTgtFamily f πX)
      (Sigma.ι (functorObjTgtFamily f πX) { i := i, t := t, b := b, w := w } :
        functorObjTgtFamily f πX { i := i, t := t, b := b, w := w : FunctorObjIndex f πX } ⟶ ∐ functorObjTgtFamily f πX)
      (CategoryStruct.comp (Y :=
          pushout (X := ∐ functorObjSrcFamily f πX) (Z := ∐ functorObjTgtFamily f πX)
            (functorObjTop f πX : ∐ functorObjSrcFamily f πX ⟶ X)
            (functorObjLeft f πX : ∐ functorObjSrcFamily f πX ⟶ ∐ functorObjTgtFamily f πX))
          (pushout.inr (functorObjTop f πX) (functorObjLeft f πX) :
            ∐ functorObjTgtFamily f πX ⟶
              pushout (X := ∐ functorObjSrcFamily f πX) (Z := ∐ functorObjTgtFamily f πX)
                (functorObjTop f πX : ∐ functorObjSrcFamily f πX ⟶ X)
                (functorObjLeft f πX : ∐ functorObjSrcFamily f πX ⟶ ∐ functorObjTgtFamily f πX))
          (πFunctorObj f πX ≫ Arrow.Hom.right τ : functorObj f πX ⟶ (Arrow.mk πY).right) :
        ∐ functorObjTgtFamily f πX ⟶ T))

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

Premises used: CategoryTheory.SmallObject.functorMap, CategoryTheory.SmallObject.ι_functorMapTgt_assoc, rfl