-- Module: Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
F : Functor.{v, v', u, u'} C D
inst_2 : PreservesColimitsOfShape (Discrete WalkingPair) F
inst_3 : PreservesColimitsOfShape (Discrete PEmpty.{1}) F
inst_4 : HasFiniteCoproducts.{v, u} C
n : ℕ
this_1 : ∀ (f : Fin n → C), PreservesColimit (Discrete.functor f) F
f : Fin (n + 1) → C
this : IsColimit.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
  Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
  (extendCofan
    (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
      F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
    (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
      (F.map coprod.inl) (F.map coprod.inr))) :=
  extendCofanIsColimit (fun (i : Fin (n + 1)) ↦ F.obj (f i))
    (isColimitOfHasCoproductOfPreservesColimit F fun (i : Fin n) ↦ f i.succ)
    (isColimitOfHasBinaryCoproductOfPreservesColimit F (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ))
j : Fin (n + 1)
i : Fin n
h :
  Eq.{v' + 1} (α :=
    Functor.obj.{0, v', 0, u'} (Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
        { as := i.castSucc : Discrete (Fin (n + 1)) } ⟶
      Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F := Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
        (Cofan.mk
          (F.obj
            (Cocone.pt.{0, v, 0, u} (F :=
              pair (f 0)
                (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                  (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
              (colimit.cocone
                (pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
          fun (j : Fin (n + 1)) ↦
          F.map
            (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
              f x ⟶
                Cocone.pt.{0, v, 0, u} (F :=
                  pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                  (colimit.cocone
                    (pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
              (BinaryCofan.inl
                (colimit.cocone
                  (pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
              (fun (i : Fin n) ↦
                NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
                    { as := i : Discrete (Fin n) } ≫
                  BinaryCofan.inr
                    (colimit.cocone
                      (pair (f 0)
                        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                          Discrete.functor fun (i : Fin n) ↦ f i.succ)
                          (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
              j)))
    (CategoryStruct.comp (Y :=
        Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
          Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
          (extendCofan
            (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
              F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
            (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
              (F.map coprod.inl) (F.map coprod.inr))))
        (NatTrans.app
            (Cocone.ι
              (extendCofan
                (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
                  F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
                (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
                  (F.map coprod.inl) (F.map coprod.inr))))
            { as := i.castSucc } :
          Functor.obj.{0, v', 0, u'} (Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
              { as := i.castSucc : Discrete (Fin (n + 1)) } ⟶
            Functor.obj.{0, v', 0, u'}
              (Functor.obj (Functor.const (Discrete (Fin (n + 1))))
                (Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
                  Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
                  (extendCofan
                    (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
                      F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
                    (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
                      (F.map coprod.inl) (F.map coprod.inr)))))
              { as := i.castSucc : Discrete (Fin (n + 1)) })
        (Iso.hom
          (Iso.refl
            (Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
              Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
              (extendCofan
                (Cofan.mk (F.obj (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)) fun (j : Fin n) ↦
                  F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) j))
                (BinaryCofan.mk (P := F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
                  (F.map coprod.inl) (F.map coprod.inr)))))) :
      Functor.obj.{0, v', 0, u'} (Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
          { as := i.castSucc : Discrete (Fin (n + 1)) } ⟶
        Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
          Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
          (Cofan.mk
            (F.obj
              (Cocone.pt.{0, v, 0, u} (F :=
                pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                (colimit.cocone
                  (pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
            fun (j : Fin (n + 1)) ↦
            F.map
              (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
                f x ⟶
                  Cocone.pt.{0, v, 0, u} (F :=
                    pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                    (colimit.cocone
                      (pair (f 0)
                        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                          Discrete.functor fun (i : Fin n) ↦ f i.succ)
                          (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                (BinaryCofan.inl
                  (colimit.cocone
                    (pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                (fun (i : Fin n) ↦
                  NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
                      { as := i : Discrete (Fin n) } ≫
                    BinaryCofan.inr
                      (colimit.cocone
                        (pair (f 0)
                          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                            Discrete.functor fun (i : Fin n) ↦ f i.succ)
                            (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                j)))
    (NatTrans.app
        (Cocone.ι
          (Cofan.mk
            (F.obj
              (Cocone.pt.{0, v, 0, u} (F :=
                pair (f 0)
                  (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                    (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                (colimit.cocone
                  (pair (f 0)
                    (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                      (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
            fun (j : Fin (n + 1)) ↦
            F.map
              (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
                f x ⟶
                  Cocone.pt.{0, v, 0, u} (F :=
                    pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                    (colimit.cocone
                      (pair (f 0)
                        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                          Discrete.functor fun (i : Fin n) ↦ f i.succ)
                          (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                (BinaryCofan.inl
                  (colimit.cocone
                    (pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                (fun (i : Fin n) ↦
                  NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))) { as := i } ≫
                    BinaryCofan.inr
                      (colimit.cocone
                        (pair (f 0)
                          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                            Discrete.functor fun (i : Fin n) ↦ f i.succ)
                            (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                j)))
        { as := i.castSucc } :
      Functor.obj.{0, v', 0, u'} (Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
          { as := i.castSucc : Discrete (Fin (n + 1)) } ⟶
        Functor.obj.{0, v', 0, u'}
          (Functor.obj (Functor.const (Discrete (Fin (n + 1))))
            (Cocone.pt.{0, v', 0, u'} (J := Discrete (Fin (n + 1))) (F :=
              Discrete.functor fun (i : Fin (n + 1)) ↦ F.obj (f i))
              (Cofan.mk
                (F.obj
                  (Cocone.pt.{0, v, 0, u} (F :=
                    pair (f 0)
                      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (i : Fin n) ↦ f i.succ)
                        (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                    (colimit.cocone
                      (pair (f 0)
                        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                          Discrete.functor fun (i : Fin n) ↦ f i.succ)
                          (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))))
                fun (j : Fin (n + 1)) ↦
                F.map
                  (Fin.cases (motive := fun (x : Fin (n + 1)) ↦
                    f x ⟶
                      Cocone.pt.{0, v, 0, u} (F :=
                        pair (f 0)
                          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                            Discrete.functor fun (i : Fin n) ↦ f i.succ)
                            (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))
                        (colimit.cocone
                          (pair (f 0)
                            (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                              Discrete.functor fun (i : Fin n) ↦ f i.succ)
                              (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                    (BinaryCofan.inl
                      (colimit.cocone
                        (pair (f 0)
                          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                            Discrete.functor fun (i : Fin n) ↦ f i.succ)
                            (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                    (fun (i : Fin n) ↦
                      NatTrans.app (Cocone.ι (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
                          { as := i : Discrete (Fin n) } ≫
                        BinaryCofan.inr
                          (colimit.cocone
                            (pair (f 0)
                              (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
                                Discrete.functor fun (i : Fin n) ↦ f i.succ)
                                (colimit.cocone (Discrete.functor fun (i : Fin n) ↦ f i.succ))))))
                    j))))
          { as := i.castSucc : Discrete (Fin (n + 1)) })
refine_2 :
  Eq (α :=
    F.obj (f i.succ) ⟶
      F.obj
        (colimit.{0, 0, v, u}
          (pair (f 0) (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))
    (F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) i) ≫ F.map coprod.inr :
      F.obj (f i.succ) ⟶ F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
    (F.map
        (CategoryStruct.comp (Y :=
          colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ))
          (colimit.ι (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := i } :
            Quiver.Hom.{v, u} (V := C)
              (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := i : Discrete (Fin n) })
              (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
          (colimit.ι
              (pair (f 0) (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
              { as := WalkingPair.right } :
            Functor.obj.{0, v, 0, u}
                (pair (f 0)
                  (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
                { as := WalkingPair.right : Discrete WalkingPair } ⟶
              colimit.{0, 0, v, u}
                (pair (f 0)
                  (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ))))) :
      F.obj (f i.succ) ⟶
        F.obj
          (colimit.{0, 0, v, u}
            (pair (f 0) (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))
refine_2₁ refine_2₂ :
  Eq (α :=
    F.obj (f i.succ) ⟶
      F.obj
        (colimit.{0, 0, v, u}
          (pair (f 0) (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))
    (F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) i ≫ coprod.inr) :
      F.obj (f i.succ) ⟶ F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ)))
    (F.map
        (CategoryStruct.comp (Y :=
          colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ))
          (colimit.ι (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := i } :
            Quiver.Hom.{v, u} (V := C)
              (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := i : Discrete (Fin n) })
              (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
          (colimit.ι
              (pair (f 0) (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
              { as := WalkingPair.right } :
            Functor.obj.{0, v, 0, u}
                (pair (f 0)
                  (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
                { as := WalkingPair.right : Discrete WalkingPair } ⟶
              colimit.{0, 0, v, u}
                (pair (f 0)
                  (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ))))) :
      F.obj (f i.succ) ⟶
        F.obj
          (colimit.{0, 0, v, u}
            (pair (f 0) (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))
⊢ (F.map (Sigma.ι.{0, v, u} (C := C) (fun (i : Fin n) ↦ f i.succ) i) ≫ F.map coprod.inr) ≫
      𝟙 (F.obj (coprod.{v, u} (f 0) (sigmaObj.{0, v, u} fun (i : Fin n) ↦ f i.succ))) =
    F.map
      (CategoryStruct.comp (Y :=
        colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ))
        (colimit.ι (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := i } :
          Quiver.Hom.{v, u} (V := C)
            (Functor.obj.{0, v, 0, u} (Discrete.functor fun (i : Fin n) ↦ f i.succ) { as := i : Discrete (Fin n) })
            (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
        (colimit.ι
            (pair (f 0) (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
            { as := WalkingPair.right } :
          Functor.obj.{0, v, 0, u}
              (pair (f 0) (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))
              { as := WalkingPair.right : Discrete WalkingPair } ⟶
            colimit.{0, 0, v, u}
              (pair (f 0)
                (colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (i : Fin n) ↦ f i.succ)))))

Tactic:
rw [comp_id, ← F.map_comp]

Premises used: CategoryTheory.Category.comp_id, CategoryTheory.Functor.map_comp