instance preservesLimit_diagramFunctor
    (X : C) (K : Type max v u) [SmallCategory K] [HasLimitsOfShape K D] (F : K ⥤ Cᵒᵖ ⥤ D) :
    PreservesLimit F (J.diagramFunctor D X) :=
  preservesLimitOfEvaluation _ _ fun W =>
    preservesLimitOfPreservesLimitCone (limit.isLimit _)
      { lift := fun E => liftToDiagramLimitObj.{w, v, u} F E
        fac := by
          intro E k
          dsimp [diagramNatTrans]
          refine' Multiequalizer.hom_ext _ _ _ (fun a => _)
          simp only [Multiequalizer.lift_ι, Multiequalizer.lift_ι_assoc, Category.assoc]
          change (_ ≫ _) ≫ _ = _
          dsimp [evaluateCombinedCones]
          erw [Category.comp_id, Category.assoc, ← NatTrans.comp_app, limit.lift_π, limit.lift_π]
          rfl
        uniq := by
          intro E m hm
          refine' Multiequalizer.hom_ext _ _ _ (fun a => limit_obj_ext (fun j => _))
          delta liftToDiagramLimitObj
          

Proof state:
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝³ : Category.{max v u, w} D
inst✝² : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)
X : C
K : Type (max v u)
inst✝¹ : SmallCategory K
inst✝ : HasLimitsOfShape K D
F : K ⥤ Cᵒᵖ ⥤ D
W : (Cover J X)ᵒᵖ
E : Cone (F ⋙ diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W)
m : E.pt ⟶ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).pt
hm :
  ∀ (j : K),
    m ≫ ((diagramFunctor J D X ⋙ (evaluation (Cover J X)ᵒᵖ D).obj W).mapCone (limit.cone F)).π.app j = E.π.app j
a : (Cover.index W.unop (limit.cone F).pt).L
j : K
⊢ (m ≫ Multiequalizer.ι (Cover.index W.unop (limit.cone F).pt) a) ≫ (limit.π F j).app (op a.Y) =
    ((fun E =>
            Multiequalizer.lift (Cover.index W.unop (limit F)) E.pt
              (fun i =>
                IsLimit.lift (isLimitOfPreserves ((evaluation Cᵒᵖ D).obj (op i.Y)) (limit.isLimit F))
                  (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation i E))
              (_ :
                ∀ (i : (Cover.index W.unop (limit F)).R),
                  ((IsLimit.liftConeMorphism
                            (IsLimit.ofIsoLimit
                              (limit.isLimit
                                (F ⋙
                                  (evaluation Cᵒᵖ D).obj
                                    (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y)))
                              (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k))
                                  (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y)).symm)
                            (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation
                              (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i) E)).hom ≫
                        ((Cones.functoriality F
                                  ((evaluation Cᵒᵖ D).obj
                                    (op (MulticospanIndex.fstTo (Cover.index W.unop (limit F)) i).Y))).mapIso
                              (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))
                                (limit.isLimit F))).hom.hom) ≫
                      MulticospanIndex.fst (Cover.index W.unop (limit F)) i =
                    ((IsLimit.liftConeMorphism
                            (IsLimit.ofIsoLimit
                              (limit.isLimit
                                (F ⋙
                                  (evaluation Cᵒᵖ D).obj
                                    (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y)))
                              (evaluateCombinedCones F (fun k => getLimitCone ((Functor.flip F).obj k))
                                  (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y)).symm)
                            (coneCompEvaluationOfConeCompDiagramFunctorCompEvaluation
                              (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i) E)).hom ≫
                        ((Cones.functoriality F
                                  ((evaluation Cᵒᵖ D).obj
                                    (op (MulticospanIndex.sndTo (Cover.index W.unop (limit F)) i).Y))).mapIso
                              (IsLimit.uniqueUpToIso (combinedIsLimit F fun k => getLimitCone ((Functor.flip F).obj k))
                                (limit.isLimit F))).hom.hom) ≫
                      MulticospanIndex.snd (Cover.index W.unop (limit F)) i))
          E ≫
        Multiequalizer.ι (Cover.index W.unop (limit.cone F).pt) a) ≫
      (limit.π F j).app (op a.Y)

Next tactic:
erw [Multiequalizer.lift_ι, Category.assoc]