Module: Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone ((diagram U).op ⋙ F)
Y Z : WalkingParallelPair
f : Y ⟶ Z
⊢ ((Functor.const WalkingParallelPair).obj c.pt).map f ≫
      WalkingParallelPair.casesOn Z (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) =
    WalkingParallelPair.casesOn Y (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) ≫
      (SheafConditionEqualizerProducts.diagram F U).map f

Tactic:
cases Y <;> cases Z <;> cases f

Goal state after:
case zero.zero.id
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone ((diagram U).op ⋙ F)
⊢ ((Functor.const WalkingParallelPair).obj c.pt).map (WalkingParallelPairHom.id WalkingParallelPair.zero) ≫
      WalkingParallelPair.casesOn WalkingParallelPair.zero (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) =
    WalkingParallelPair.casesOn WalkingParallelPair.zero (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) ≫
      (SheafConditionEqualizerProducts.diagram F U).map (WalkingParallelPairHom.id WalkingParallelPair.zero)

case zero.one.left
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone ((diagram U).op ⋙ F)
⊢ ((Functor.const WalkingParallelPair).obj c.pt).map WalkingParallelPairHom.left ≫
      WalkingParallelPair.casesOn WalkingParallelPair.one (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) =
    WalkingParallelPair.casesOn WalkingParallelPair.zero (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) ≫
      (SheafConditionEqualizerProducts.diagram F U).map WalkingParallelPairHom.left

case zero.one.right
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone ((diagram U).op ⋙ F)
⊢ ((Functor.const WalkingParallelPair).obj c.pt).map WalkingParallelPairHom.right ≫
      WalkingParallelPair.casesOn WalkingParallelPair.one (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) =
    WalkingParallelPair.casesOn WalkingParallelPair.zero (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) ≫
      (SheafConditionEqualizerProducts.diagram F U).map WalkingParallelPairHom.right

case one.one.id
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c : Cone ((diagram U).op ⋙ F)
⊢ ((Functor.const WalkingParallelPair).obj c.pt).map (WalkingParallelPairHom.id WalkingParallelPair.one) ≫
      WalkingParallelPair.casesOn WalkingParallelPair.one (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) =
    WalkingParallelPair.casesOn WalkingParallelPair.one (Pi.lift fun i ↦ c.π.app (op (single i)))
        (Pi.lift fun b ↦ c.π.app (op (Pairwise.pair b.1 b.2))) ≫
      (SheafConditionEqualizerProducts.diagram F U).map (WalkingParallelPairHom.id WalkingParallelPair.one)