-- Module: Mathlib.CategoryTheory.Sites.LocallySurjective
Goal:
C : Type u
inst : Category.{v, u} C
S : C
ι : Type u_2
inst_1 : Small.{w, u_2} ι
X : ι → C
f : (i : ι) → X i ⟶ S
inst_2 : LocallySmall.{w, v, u} C
c : Cofan (C := Functor.{v, w, u, w + 1} Cᵒᵖ (Type w)) fun (i : ι) ↦ obj shrinkYoneda.{w, v, u} (X i)
hc : IsColimit c
U : C
g : U ⟶ S
V : C
v : V ⟶ U
i : ι
a : V ⟶ X i
hw :
  Eq (α := (fun (X : Type w) ↦ X) (obj.{v, w, u, w + 1} (obj shrinkYoneda.{w, v, u} S) (op V)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app (Cofan.IsColimit.desc hc fun (i : ι) ↦ map shrinkYoneda.{w, v, u} (f i)) (op V)) :
          (a : (fun (X : Type w) ↦ X) (c.pt.obj (op V))) →
            (fun (X : Type w) ↦ X) ((shrinkYoneda.{w, v, u}.obj S).obj (op V)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app (Cocone.ι (mapCocone (obj (evaluation Cᵒᵖ (Type w)) (op V)) c))
                  { as := i : Discrete ι }) :
              (a :
                  (fun (X : Type w) ↦ X)
                    (((Discrete.functor fun (i : ι) ↦ shrinkYoneda.{w, v, u}.obj (X i)) ⋙
                          (evaluation Cᵒᵖ (Type w)).obj (op V)).obj
                      { as := i })) →
                (fun (X : Type w) ↦ X)
                  (((const (Discrete ι)).obj (((evaluation Cᵒᵖ (Type w)).obj (op V)).mapCocone c).pt).obj { as := i }))
            ((Equiv.symm shrinkYonedaObjObjEquiv :
                  (a : unop (op V) ⟶ X { as := i }.as) → (shrinkYoneda.{w, v, u}.obj (X { as := i }.as)).obj (op V))
                a :
              obj.{v, w, u, w + 1} (obj shrinkYoneda.{w, v, u} (X { as := i : Discrete ι }.as)) (op V)) :
          (fun (X : Type w) ↦ X)
            (obj
              (obj (const (Discrete ι))
                (Cocone.pt (F :=
                  comp (D := Functor.{v, w, u, w + 1} Cᵒᵖ (Type w))
                    (Discrete.functor fun (i : ι) ↦ obj shrinkYoneda.{w, v, u} (X i))
                    (obj (evaluation Cᵒᵖ (Type w)) (op V)))
                  (mapCocone (obj (evaluation Cᵒᵖ (Type w)) (op V)) c)))
              { as := i : Discrete ι })) :
      (fun (X : Type w) ↦ X) (obj.{v, w, u, w + 1} (obj shrinkYoneda.{w, v, u} S) (op V)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (map (obj shrinkYoneda.{w, v, u} S) v.op) :
          (a : (fun (X : Type w) ↦ X) ((shrinkYoneda.{w, v, u}.obj S).obj (op U))) →
            (fun (X : Type w) ↦ X) ((shrinkYoneda.{w, v, u}.obj S).obj (op V)))
        ((Equiv.symm shrinkYonedaObjObjEquiv : (a : U ⟶ S) → (shrinkYoneda.{w, v, u}.obj S).obj (op U)) g :
          obj.{v, w, u, w + 1} (obj shrinkYoneda.{w, v, u} S) (op U)) :
      (fun (X : Type w) ↦ X) (obj.{v, w, u, w + 1} (obj shrinkYoneda.{w, v, u} S) (op V)))
h :
  Eq.{w + 1} (α := obj.{v, w, u, w + 1} (obj shrinkYoneda.{w, v, u} S) (op V))
    ((Equiv.symm shrinkYonedaObjObjEquiv : (a : unop (op V) ⟶ S) → (shrinkYoneda.{w, v, u}.obj S).obj (op V))
        (a ≫ f i) :
      obj.{v, w, u, w + 1} (obj shrinkYoneda.{w, v, u} S) (op V))
    ((Equiv.symm shrinkYonedaObjObjEquiv : (a : unop (op V) ⟶ S) → (shrinkYoneda.{w, v, u}.obj S).obj (op V)) (v ≫ g) :
      obj.{v, w, u, w + 1} (obj shrinkYoneda.{w, v, u} S) (op V))
⊢ ∃ (Y : C) (h : V ⟶ Y) (g_1 : Y ⟶ S), Presieve.ofArrows X f g_1 ∧ h ≫ g_1 = v ≫ g

Tactic:
refine ⟨_, a, _, ⟨i⟩, shrinkYonedaObjObjEquiv.symm.injective ?_⟩