@[reassoc (attr := simp)]
theorem plusCompIso_whiskerRight {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :
    whiskerRight (J.plusMap η) F ≫ (J.plusCompIso F Q).hom =
      (J.plusCompIso F P).hom ≫ J.plusMap (whiskerRight η F) := by
  ext X
  apply (isColimitOfPreserves F (colimit.isColimit (J.diagram P X.unop))).hom_ext
  intro W
  dsimp [plusObj, plusMap]
  simp only [ι_colimMap, whiskerRight_app, ι_plusCompIso_hom_assoc,
    GrothendieckTopology.diagramNatTrans_app]
  simp only [← Category.assoc, ← F.map_comp]
  dsimp [colimMap, IsColimit.map]
  simp only [colimit.ι_desc]
  dsimp [Cocones.precompose]
  simp only [Functor.map_comp, Category.assoc, ι_plusCompIso_hom]
  simp only [← Category.assoc]
  congr 1
  -- porting note: this used to work with `ext`
  -- See https://github.com/leanprover-community/mathlib4/issues/5229
  apply Multiequalizer.hom_ext
  intro a
  dsimp
  

Proof state:
case w.h.e_a.h
C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D
inst✝⁴ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E
inst✝³ : (X : C) → (W : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index W P)) F
P✝ : Cᵒᵖ ⥤ D
inst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ E
inst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ F
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (Cover J X.unop)ᵒᵖ
a : (Cover.index W.unop (Q ⋙ F)).L
⊢ (F.map
          (Multiequalizer.lift (Cover.index W.unop Q) (multiequalizer (Cover.index W.unop P))
            (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app (op i.Y))
            (_ :
              ∀ (i : (Cover.index W.unop Q).R),
                (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app (op i.Y))
                      (MulticospanIndex.fstTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.fst (Cover.index W.unop Q) i =
                  (fun i => Multiequalizer.ι (Cover.index W.unop P) i ≫ η.app (op i.Y))
                      (MulticospanIndex.sndTo (Cover.index W.unop Q) i) ≫
                    MulticospanIndex.snd (Cover.index W.unop Q) i)) ≫
        (diagramCompIso J F Q X.unop).hom.app W) ≫
      Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a =
    ((diagramCompIso J F P X.unop).hom.app W ≫
        Multiequalizer.lift (Cover.index W.unop (Q ⋙ F)) (multiequalizer (Cover.index W.unop (P ⋙ F)))
          (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ F.map (η.app (op i.Y)))
          (_ :
            ∀ (i : (Cover.index W.unop (Q ⋙ F)).R),
              (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app (op i.Y))
                    (MulticospanIndex.fstTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.fst (Cover.index W.unop (Q ⋙ F)) i =
                (fun i => Multiequalizer.ι (Cover.index W.unop (P ⋙ F)) i ≫ (whiskerRight η F).app (op i.Y))
                    (MulticospanIndex.sndTo (Cover.index W.unop (Q ⋙ F)) i) ≫
                  MulticospanIndex.snd (Cover.index W.unop (Q ⋙ F)) i)) ≫
      Multiequalizer.ι (Cover.index W.unop (Q ⋙ F)) a

Next tactic:
simp only [diagramCompIso_hom_ι_assoc, Multiequalizer.lift_ι, diagramCompIso_hom_ι,
    Category.assoc]