-- Mathlib/CategoryTheory/Limits/Preserves/Presheaf.lean
theorem iso_hom [IsFiltered (CostructuredArrow yoneda A)] : (iso A K).hom = limit.post K A

Proof state:
C : Type u
inst✝⁴ : SmallCategory C
inst✝³ : HasFiniteColimits C
A : Cᵒᵖ ⥤ Type u
J : Type
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K : J ⥤ Cᵒᵖ
inst✝ : IsFiltered (CostructuredArrow yoneda A)
j : J
i : CostructuredArrow yoneda A
⊢ colimit.ι ((CostructuredArrow.proj yoneda A ⋙ yoneda) ⋙ (evaluation Cᵒᵖ (Type u)).obj (limit K)) i ≫
      (colimitObjIsoColimitCompEvaluation (CostructuredArrow.proj yoneda A ⋙ yoneda) (limit K)).inv ≫
        (((Presheaf.isColimitTautologicalCocone A).coconePointUniqueUpToIso
                  (colimit.isColimit (CostructuredArrow.proj yoneda A ⋙ yoneda))).app
              (limit K)).inv ≫
          limit.post K A ≫ limit.π (K ⋙ A) j =
    colimit.ι ((CostructuredArrow.proj yoneda A ⋙ yoneda) ⋙ (evaluation Cᵒᵖ (Type u)).obj (limit K)) i ≫
      (HasColimit.isoOfNatIso
            ((CostructuredArrow.proj yoneda A).associator yoneda ((evaluation Cᵒᵖ (Type u)).obj (limit K)))).hom ≫
        (HasColimit.isoOfNatIso (isoAux A K)).hom ≫
          (HasColimit.isoOfNatIso
                (preservesLimitIso
                  (coyoneda ⋙
                    (whiskeringLeft (CostructuredArrow yoneda A) C (Type u)).obj (CostructuredArrow.proj yoneda A))
                  K)).hom ≫
            (HasColimit.isoOfNatIso (HasLimit.isoOfNatIso (functorToInterchangeIso A K).symm)).hom ≫
              (colimitLimitIso (functorToInterchange A K)).hom ≫
                (HasLimit.isoOfNatIso (HasColimit.isoOfNatIso (flipFunctorToInterchange A K))).hom ≫
                  (HasLimit.isoOfNatIso
                        (colimitCompWhiskeringLeftIsoCompColimit (CostructuredArrow.proj yoneda A ⋙ yoneda) K)).hom ≫
                    (HasLimit.isoOfNatIso
                          (isoWhiskerLeft K
                            ((colimit.isColimit (CostructuredArrow.proj yoneda A ⋙ yoneda)).coconePointUniqueUpToIso
                              (Presheaf.isColimitTautologicalCocone A)))).hom ≫
                      limit.π (K ⋙ A) j

Tactic:
rw [HasLimit.isoOfNatIso_hom_π, HasLimit.isoOfNatIso_hom_π_assoc, limit.post_π,
  colimitObjIsoColimitCompEvaluation_ι_inv_assoc (CostructuredArrow.proj yoneda A ⋙ yoneda),
  Iso.app_inv, ← NatTrans.comp_app_assoc, colimit.comp_coconePointUniqueUpToIso_inv,
  Presheaf.tautologicalCocone_ι_app, HasColimit.isoOfNatIso_ι_hom_assoc,
  HasLimit.isoOfNatIso_hom_π_assoc, HasColimit.isoOfNatIso_ι_hom_assoc,
  HasColimit.isoOfNatIso_ι_hom_assoc, HasColimit.isoOfNatIso_ι_hom_assoc,
  ι_colimitLimitIso_limit_π_assoc, isoAux_hom_app, ← NatTrans.comp_app_assoc,
  ← NatTrans.comp_app_assoc, Category.assoc, HasLimit.isoOfNatIso_hom_π,
  preservesLimitIso_hom_π_assoc, Iso.symm_hom,
  ← NatTrans.comp_app_assoc, HasColimit.isoOfNatIso_ι_hom,
  ← NatTrans.comp_app_assoc, Category.assoc,
  ι_colimitCompWhiskeringLeftIsoCompColimit_hom,
  NatTrans.comp_app, Category.assoc, isoWhiskerLeft_hom, NatTrans.comp_app, Category.assoc,
  ← NatTrans.comp_app, ← whiskerLeft_comp, colimit.comp_coconePointUniqueUpToIso_hom]

Result:
C : Type u
inst✝⁴ : SmallCategory C
inst✝³ : HasFiniteColimits C
A : Cᵒᵖ ⥤ Type u
J : Type
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
K : J ⥤ Cᵒᵖ
inst✝ : IsFiltered (CostructuredArrow yoneda A)
j : J
i : CostructuredArrow yoneda A
⊢ i.hom.app (limit K) ≫ A.map (limit.π K j) =
    ((CostructuredArrow.proj yoneda A).associator yoneda ((evaluation Cᵒᵖ (Type u)).obj (limit K))).hom.app i ≫
      (fun X => 𝟙 (Opposite.unop (limit K) ⟶ X.left)) i ≫
        ((coyoneda ⋙ (whiskeringLeft (CostructuredArrow yoneda A) C (Type u)).obj (CostructuredArrow.proj yoneda A)).map
                (limit.π K j)).app
            i ≫
          ((functorToInterchangeIso A K).inv.app j).app i ≫
            ((flipFunctorToInterchange A K).hom.app i).app j ≫
              (whiskerLeft K ((Presheaf.tautologicalCocone A).ι.app i)).app j