Theorem: CategoryTheory.Limits.colimitCoyonedaHomIsoLimit'_π_apply
File: Mathlib/CategoryTheory/Limits/IndYoneda.lean

Proof steps:
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ limit.π (D ⋙ F ⋙ uliftFunctor.{u₁, u₂}) i ((colimitCoyonedaHomIsoLimit' D F).hom f) =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
state_after: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ ((colimitCoyonedaHomIsoLimit' D F).hom ≫ limit.π (D ⋙ F ⋙ uliftFunctor.{u₁, u₂}) i) f =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
tactic: change ((colimitCoyonedaHomIsoLimit' D F).hom ≫ (limit.π (D ⋙ F ⋙ uliftFunctor.{u₁}) i)) f = _
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ ((colimitCoyonedaHomIsoLimit' D F).hom ≫ limit.π (D ⋙ F ⋙ uliftFunctor.{u₁, u₂}) i) f =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
state_after: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ ((colimitHomIsoLimitYoneda' (D.op ⋙ coyoneda) F).hom ≫
        limit.π ((D.op ⋙ coyoneda).rightOp ⋙ yoneda.obj F) i ≫
          (isoWhiskerLeft (D ⋙ Prod.sectl C F) (coyonedaLemma C)).hom.app i)
      f =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
tactic: simp only [colimitCoyonedaHomIsoLimit', Iso.trans_hom, Category.assoc, HasLimit.isoOfNatIso_hom_π]
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ ((colimitHomIsoLimitYoneda' (D.op ⋙ coyoneda) F).hom ≫
        limit.π ((D.op ⋙ coyoneda).rightOp ⋙ yoneda.obj F) i ≫
          (isoWhiskerLeft (D ⋙ Prod.sectl C F) (coyonedaLemma C)).hom.app i)
      f =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
state_after: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ ((coyoneda.map (colimit.ι (D.op ⋙ coyoneda) (op i)).op).app F ≫
        (isoWhiskerLeft (D ⋙ Prod.sectl C F) (coyonedaLemma C)).hom.app i)
      f =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
tactic: rw [← Category.assoc, colimitHomIsoLimitYoneda'_hom_comp_π]
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ ((coyoneda.map (colimit.ι (D.op ⋙ coyoneda) (op i)).op).app F ≫
        (isoWhiskerLeft (D ⋙ Prod.sectl C F) (coyonedaLemma C)).hom.app i)
      f =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
state_after: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ Equiv.ulift.symm (coyonedaEquiv (colimit.ι (D.op ⋙ coyoneda) (op i) ≫ f)) =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
tactic: dsimp [coyonedaLemma]
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ Equiv.ulift.symm (coyonedaEquiv (colimit.ι (D.op ⋙ coyoneda) (op i) ≫ f)) =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
state_after: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ Equiv.ulift.symm (f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i)))) =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
tactic: erw [coyonedaEquiv_comp, coyonedaEquiv_apply]
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
D : I ⥤ C
F : C ⥤ Type u₂
inst✝¹ : HasColimit (D.op ⋙ coyoneda)
inst✝ : HasLimitsOfShape I (Type (max u₁ u₂))
f : colimit (D.op ⋙ coyoneda) ⟶ F
i : I
⊢ Equiv.ulift.symm (f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i)))) =
    { down := f.app (D.obj i) ((colimit.ι (D.op ⋙ coyoneda) (op i)).app (D.obj i) (𝟙 (D.obj i))) }
state_after: no goals
tactic: rfl