/-- If `F` preserves the initial object and binary coproducts, then it preserves products indexed by
`Fin n` for any `n`.
-/
noncomputable def preservesFinOfPreservesBinaryAndInitial :
    ∀ (n : ℕ) (f : Fin n → C), PreservesColimit (Discrete.functor f) F
  | 0 => fun f => by
    letI : PreservesColimitsOfShape (Discrete (Fin 0)) F :=
      preservesColimitsOfShapeOfEquiv.{0, 0} (Discrete.equivalence finZeroEquiv'.symm) _
    infer_instance
  | n + 1 => by
    haveI := preservesFinOfPreservesBinaryAndInitial n
    intro f
    refine'
      preservesColimitOfPreservesColimitCocone
        (extendCofanIsColimit f (colimit.isColimit _) (colimit.isColimit _)) _
    apply (isColimitMapCoconeCofanMkEquiv _ _ _).symm _
    let this :=
      extendCofanIsColimit (fun i => F.obj (f i))
        (isColimitOfHasCoproductOfPreservesColimit F _)
        (isColimitOfHasBinaryCoproductOfPreservesColimit F _ _)
    refine' IsColimit.ofIsoColimit this _
    apply Cocones.ext _ _
    apply Iso.refl _
    rintro ⟨j⟩
    refine' Fin.inductionOn j ?_ ?_
    · apply Category.comp_id
    · rintro i _
      

Proof state:
case mk.refine'_2
J : Type v
inst✝⁵ : SmallCategory J
C : Type u
inst✝⁴ : Category.{v, u} C
D : Type u'
inst✝³ : Category.{v', u'} D
F : C ⥤ D
inst✝² : PreservesColimitsOfShape (Discrete WalkingPair) F
inst✝¹ : PreservesColimitsOfShape (Discrete PEmpty.{1}) F
inst✝ : HasFiniteCoproducts C
n : ℕ
this✝ : (f : Fin n → C) → PreservesColimit (Discrete.functor f) F
f : Fin (n + 1) → C
this : IsColimit
  (extendCofan (Cofan.mk (F.obj (∐ fun i => f (Fin.succ i))) fun j => F.map (Sigma.ι (fun i => f (Fin.succ i)) j))
    (BinaryCofan.mk (F.map coprod.inl) (F.map coprod.inr))) :=
  extendCofanIsColimit (fun i => F.obj (f i)) (isColimitOfHasCoproductOfPreservesColimit F fun i => f (Fin.succ i))
    (isColimitOfHasBinaryCoproductOfPreservesColimit F (f 0) (∐ fun i => f (Fin.succ i)))
j : Fin (n + 1)
i : Fin n
a✝ :
  (extendCofan (Cofan.mk (F.obj (∐ fun i => f (Fin.succ i))) fun j => F.map (Sigma.ι (fun i => f (Fin.succ i)) j))
              (BinaryCofan.mk (F.map coprod.inl) (F.map coprod.inr))).ι.app
        { as := Fin.castSucc i } ≫
      (Iso.refl
          (extendCofan
              (Cofan.mk (F.obj (∐ fun i => f (Fin.succ i))) fun j => F.map (Sigma.ι (fun i => f (Fin.succ i)) j))
              (BinaryCofan.mk (F.map coprod.inl) (F.map coprod.inr))).pt).hom =
    (Cofan.mk (F.obj (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f (Fin.succ i))).pt)).pt)
            fun j =>
            F.map
              (Fin.cases
                (BinaryCofan.inl
                  (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f (Fin.succ i))).pt)))
                (fun i =>
                  (colimit.cocone (Discrete.functor fun i => f (Fin.succ i))).ι.app { as := i } ≫
                    BinaryCofan.inr
                      (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f (Fin.succ i))).pt)))
                j)).ι.app
      { as := Fin.castSucc i }
⊢ (extendCofan (Cofan.mk (F.obj (∐ fun i => f (Fin.succ i))) fun j => F.map (Sigma.ι (fun i => f (Fin.succ i)) j))
              (BinaryCofan.mk (F.map coprod.inl) (F.map coprod.inr))).ι.app
        { as := Fin.succ i } ≫
      (Iso.refl
          (extendCofan
              (Cofan.mk (F.obj (∐ fun i => f (Fin.succ i))) fun j => F.map (Sigma.ι (fun i => f (Fin.succ i)) j))
              (BinaryCofan.mk (F.map coprod.inl) (F.map coprod.inr))).pt).hom =
    (Cofan.mk (F.obj (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f (Fin.succ i))).pt)).pt)
            fun j =>
            F.map
              (Fin.cases
                (BinaryCofan.inl
                  (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f (Fin.succ i))).pt)))
                (fun i =>
                  (colimit.cocone (Discrete.functor fun i => f (Fin.succ i))).ι.app { as := i } ≫
                    BinaryCofan.inr
                      (colimit.cocone (pair (f 0) (colimit.cocone (Discrete.functor fun i => f (Fin.succ i))).pt)))
                j)).ι.app
      { as := Fin.succ i }

Next tactic:
dsimp [extendCofan_ι_app, Iso.refl_hom, Cofan.mk_ι_app]