-- Module: Mathlib.CategoryTheory.Sites.Sieves
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
F : Functor.{v₁, v₂, u₁, u₂} C D
X Y Z : C
f_1 : Y ⟶ X
S R : Sieve.{v₁, u₁} X
U V : Cᵒᵖ
f : U ⟶ V
dsimp :
  Eq.{v₁ + 1} (α :=
    { x : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) U //
        S.arrows (Y := Opposite.unop U)
          ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U) → Opposite.unop U ⟶ X) x) } ⟶
      { g : Opposite.unop V ⟶ X // S.arrows g })
    (CategoryStruct.comp (Y :=
      { x : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) V //
        S.arrows (Y := Opposite.unop V)
          ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj V) → Opposite.unop V ⟶ X) x) })
      (↾fun
          (x :
            { x : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) U //
              S.arrows (Y := Opposite.unop U)
                ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U) → Opposite.unop U ⟶ X) x) }) ↦
        Subtype.mk (p := fun (x : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) V) ↦
          S.arrows (Y := Opposite.unop V)
            ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj V) → Opposite.unop V ⟶ X) x))
          ((hom (FC := TypeCat.Fun) (Functor.map (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) f) :
              (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U) → (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj V)
            (↑x : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U))
          ⋯)
      (Iso.hom
        (Equiv.toIso
          {
            toFun :=
              fun
                (a :
                  { a : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) V //
                    S.arrows (Y := Opposite.unop V)
                      ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj V) → Opposite.unop V ⟶ X)
                        a) }) ↦
              Subtype.mk (p := fun (b : Opposite.unop V ⟶ X) ↦ S.arrows b)
                ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj V) → Opposite.unop V ⟶ X)
                  (↑a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj V))
                ⋯,
            invFun := fun (b : { b : Opposite.unop V ⟶ X // S.arrows b }) ↦
              Subtype.mk (p := fun (a : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) V) ↦
                S.arrows (Y := Opposite.unop V)
                  ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj V) → Opposite.unop V ⟶ X) a))
                ((Equiv.symm shrinkYonedaObjObjEquiv :
                    (a : Opposite.unop V ⟶ X) → (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj V)
                  (↑b : Opposite.unop V ⟶ X))
                ⋯,
            left_inv := ⋯, right_inv := ⋯ })))
    (CategoryStruct.comp (Y := { g : Opposite.unop U ⟶ X // S.arrows g })
      (Iso.hom
        (Equiv.toIso
          {
            toFun :=
              fun
                (a :
                  { a : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) U //
                    S.arrows (Y := Opposite.unop U)
                      ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U) → Opposite.unop U ⟶ X)
                        a) }) ↦
              Subtype.mk (p := fun (b : Opposite.unop U ⟶ X) ↦ S.arrows b)
                ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U) → Opposite.unop U ⟶ X)
                  (↑a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U))
                ⋯,
            invFun := fun (b : { b : Opposite.unop U ⟶ X // S.arrows b }) ↦
              Subtype.mk (p := fun (a : Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) U) ↦
                S.arrows (Y := Opposite.unop U)
                  ((shrinkYonedaObjObjEquiv : (a : (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U) → Opposite.unop U ⟶ X) a))
                ((Equiv.symm shrinkYonedaObjObjEquiv :
                    (a : Opposite.unop U ⟶ X) → (shrinkYoneda.{v₁, v₁, u₁}.obj X).obj U)
                  (↑b : Opposite.unop U ⟶ X))
                ⋯,
            left_inv := ⋯, right_inv := ⋯ }))
      (↾fun (g : { g : Opposite.unop U ⟶ X // S.arrows g }) ↦
        Subtype.mk (p := fun (g : Opposite.unop V ⟶ X) ↦ S.arrows g) (f.unop ≫ (↑g : Opposite.unop U ⟶ X)) ⋯))
⊢ (Subfunctor.toFunctor (F := Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) (shrinkFunctor.{v₁, v₁, u₁} S)).map f ≫
      Iso.hom
        ((fun (Y : Cᵒᵖ) ↦
            (Equiv.subtypeEquiv (p :=
                Membership.mem (γ := Set (Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) Y))
                  (Subfunctor.obj (shrinkFunctor.{v₁, v₁, u₁} S) Y))
                (q := S.arrows) shrinkYonedaObjObjEquiv ⋯).toIso)
          V) =
    Iso.hom
        ((fun (Y : Cᵒᵖ) ↦
            (Equiv.subtypeEquiv (p :=
                Membership.mem (γ := Set (Functor.obj (Functor.obj shrinkYoneda.{v₁, v₁, u₁} X) Y))
                  (Subfunctor.obj (shrinkFunctor.{v₁, v₁, u₁} S) Y))
                (q := S.arrows) shrinkYonedaObjObjEquiv ⋯).toIso)
          U) ≫
      S.functor.map f

Tactic:
dsimp [Equiv.subtypeEquiv]

Premises used: Equiv.subtypeEquiv