/-- The diagram consisting of the `U i` and `U i ⊓ U j` is cofinal in the diagram
of all opens contained in some `U i`.
-/
instance : Functor.Final (pairwiseToOpensLeCover U) :=
  ⟨fun V =>
    isConnected_of_zigzag fun A B => by
      rcases A with ⟨⟨⟨⟩⟩, ⟨i⟩ | ⟨i, j⟩, a⟩ <;> rcases B with ⟨⟨⟨⟩⟩, ⟨i'⟩ | ⟨i', j'⟩, b⟩
      · refine'
          ⟨[{   left := ⟨⟨⟩⟩
                right := pair i i'
                hom := (le_inf a.le b.le).hom }, _], _, rfl⟩
        exact
          List.Chain.cons
            (Or.inr
              ⟨{  left := 𝟙 _
                  right := left i i' }⟩)
            (List.Chain.cons
              (Or.inl
                ⟨{  left := 𝟙 _
                    right := right i i' }⟩)
              List.Chain.nil)
      · refine'
          ⟨[{   left := ⟨⟨⟩⟩
                right := pair i' i
                hom := (le_inf (b.le.trans inf_le_left) a.le).hom },
              { left := ⟨⟨⟩⟩
                right := single i'
                hom := (b.le.trans inf_le_left).hom }, _], _, rfl⟩
        exact
          List.Chain.cons
            (Or.inr
              ⟨{  left := 𝟙 _
                  right := right i' i }⟩)
            (List.Chain.cons
              (Or.inl
                ⟨{  left := 𝟙 _
                    right := left i' i }⟩)
              (List.Chain.cons
                (Or.inr
                  ⟨{  left := 𝟙 _
                      right := left i' j' }⟩)
                List.Chain.nil))
      · refine'
          ⟨[{   left := ⟨⟨⟩⟩
                right := single i
                hom := (a.le.trans inf_le_left).hom },
              { left := ⟨⟨⟩⟩
                right := pair i i'
                hom := (le_inf (a.le.trans inf_le_left) b.le).hom }, _], _, rfl⟩
        exact
          List.Chain.cons
            (Or.inl
              ⟨{  left := 𝟙 _
                  right := left i j }⟩)
            (List.Chain.cons
              (Or.inr
                ⟨{  left := 𝟙 _
                    right := left i i' }⟩)
              (List.Chain.cons
                (Or.inl
                  ⟨{  left := 𝟙 _
                      right := right i i' }⟩)
                List.Chain.nil))
      · refine'
          ⟨[{   left := ⟨⟨⟩⟩
                right := single i
                hom := (a.le.trans inf_le_left).hom },
              { left := ⟨⟨⟩⟩
                right := pair i i'
                hom := (le_inf (a.le.trans inf_le_left) (b.le.trans inf_le_left)).hom },
              { left := ⟨⟨⟩⟩
                right := single i'
                hom := (b.le.trans inf_le_left).hom }, _], _, rfl⟩
        

Proof state:
case mk.mk.unit.pair.mk.mk.unit.pair
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
ι : Type w
U : ι → Opens ↑X
V : OpensLeCover U
i j : ι
a : (Functor.fromPUnit V).obj { as := PUnit.unit } ⟶ (pairwiseToOpensLeCover U).obj (Pairwise.pair i j)
i' j' : ι
b : (Functor.fromPUnit V).obj { as := PUnit.unit } ⟶ (pairwiseToOpensLeCover U).obj (Pairwise.pair i' j')
⊢ List.Chain Zag { left := { as := PUnit.unit }, right := Pairwise.pair i j, hom := a }
    [{ left := { as := PUnit.unit }, right := single i,
        hom := LE.le.hom (_ : ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj ≤ U i) },
      { left := { as := PUnit.unit }, right := Pairwise.pair i i',
        hom := LE.le.hom (_ : ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj ≤ U i ⊓ U i') },
      { left := { as := PUnit.unit }, right := single i',
        hom := LE.le.hom (_ : ((Functor.fromPUnit V).obj { as := PUnit.unit }).obj ≤ U i') },
      { left := { as := PUnit.unit }, right := Pairwise.pair i' j', hom := b }]

Next tactic:
exact
          List.Chain.cons
            (Or.inl
              ⟨{  left := 𝟙 _
                  right := left i j }⟩)
            (List.Chain.cons
              (Or.inr
                ⟨{  left := 𝟙 _
                    right := left i i' }⟩)
              (List.Chain.cons
                (Or.inl
                  ⟨{  left := 𝟙 _
                      right := right i i' }⟩)
                (List.Chain.cons
                  (Or.inr
                    ⟨{  left := 𝟙 _
                        right := left i' j' }⟩)
                  List.Chain.nil)))