Module: Mathlib.CategoryTheory.Presentable.CardinalDirectedPoset

Goal state before:
pp: case refine_1
κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
hJ : HasCardinalLT (↑J.obj) κ'
h : κ ≤ κ'
A : Type u
x✝¹ : SmallCategory A
x✝ : IsCardinalFiltered A κ'
F : A ⥤ CardinalDirectedPoset κ
c : Cocone F
this✝ : IsFiltered A
this : IsCardinalFiltered A κ
hc : IsColimit ((forget (CardinalDirectedPoset κ)).mapCocone c)
f : J ⟶ c.pt
j : ↑J.obj → A
g : (x : ↑J.obj) → (fun X ↦ X) ((F ⋙ forget (CardinalDirectedPoset κ)).obj (j x))
hg :
  ∀ (x : ↑J.obj),
    (ConcreteCategory.hom (((forget (CardinalDirectedPoset κ)).mapCocone c).ι.app (j x))) (g x) =
      (ConcreteCategory.hom f) x
m : A := IsCardinalFiltered.max j hJ
φ : ↑J.obj → ↑(F.obj m).obj := fun x ↦ (ConcreteCategory.hom (F.map (IsCardinalFiltered.toMax j hJ x))) (g x)
hφ : ∀ (x : ↑J.obj), (ConcreteCategory.hom f) x = (ConcreteCategory.hom (c.ι.app m)) (φ x)
⊢ ∃ i xi, f = (ConcreteCategory.hom (((coyoneda.obj (Opposite.op J)).mapCocone c).ι.app i)) xi

Tactic:
refine ⟨m,
        ObjectProperty.homMk (PartOrdEmb.ofHom
          { toFun := φ
            inj' x y h := Hom.injective f (by simpa [hφ])
            map_rel_iff' {x y} := ?_ }), ?_⟩

Goal state after:
case refine_1.refine_1
κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
hJ : HasCardinalLT (↑J.obj) κ'
h : κ ≤ κ'
A : Type u
x✝¹ : SmallCategory A
x✝ : IsCardinalFiltered A κ'
F : A ⥤ CardinalDirectedPoset κ
c : Cocone F
this✝ : IsFiltered A
this : IsCardinalFiltered A κ
hc : IsColimit ((forget (CardinalDirectedPoset κ)).mapCocone c)
f : J ⟶ c.pt
j : ↑J.obj → A
g : (x : ↑J.obj) → (fun X ↦ X) ((F ⋙ forget (CardinalDirectedPoset κ)).obj (j x))
hg :
  ∀ (x : ↑J.obj),
    (ConcreteCategory.hom (((forget (CardinalDirectedPoset κ)).mapCocone c).ι.app (j x))) (g x) =
      (ConcreteCategory.hom f) x
m : A := IsCardinalFiltered.max j hJ
φ : ↑J.obj → ↑(F.obj m).obj := fun x ↦ (ConcreteCategory.hom (F.map (IsCardinalFiltered.toMax j hJ x))) (g x)
hφ : ∀ (x : ↑J.obj), (ConcreteCategory.hom f) x = (ConcreteCategory.hom (c.ι.app m)) (φ x)
x y : ↑(Opposite.unop (Opposite.op J)).1
⊢ { toFun := φ, inj' := ⋯ } x ≤ { toFun := φ, inj' := ⋯ } y ↔ x ≤ y

case refine_1.refine_2
κ : Cardinal.{u}
inst✝¹ : Fact κ.IsRegular
J : CardinalDirectedPoset κ
κ' : Cardinal.{u}
inst✝ : Fact κ'.IsRegular
hJ : HasCardinalLT (↑J.obj) κ'
h : κ ≤ κ'
A : Type u
x✝¹ : SmallCategory A
x✝ : IsCardinalFiltered A κ'
F : A ⥤ CardinalDirectedPoset κ
c : Cocone F
this✝ : IsFiltered A
this : IsCardinalFiltered A κ
hc : IsColimit ((forget (CardinalDirectedPoset κ)).mapCocone c)
f : J ⟶ c.pt
j : ↑J.obj → A
g : (x : ↑J.obj) → (fun X ↦ X) ((F ⋙ forget (CardinalDirectedPoset κ)).obj (j x))
hg :
  ∀ (x : ↑J.obj),
    (ConcreteCategory.hom (((forget (CardinalDirectedPoset κ)).mapCocone c).ι.app (j x))) (g x) =
      (ConcreteCategory.hom f) x
m : A := IsCardinalFiltered.max j hJ
φ : ↑J.obj → ↑(F.obj m).obj := fun x ↦ (ConcreteCategory.hom (F.map (IsCardinalFiltered.toMax j hJ x))) (g x)
hφ : ∀ (x : ↑J.obj), (ConcreteCategory.hom f) x = (ConcreteCategory.hom (c.ι.app m)) (φ x)
⊢ f =
    (ConcreteCategory.hom (((coyoneda.obj (Opposite.op J)).mapCocone c).ι.app m))
      (ObjectProperty.homMk (PartOrdEmb.ofHom { toFun := φ, inj' := ⋯, map_rel_iff' := ⋯ }))