-- Mathlib/CategoryTheory/Sites/CompatiblePlus.lean
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)

Proof state:
case w.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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ F.map (colimit.ι (J.diagram P (unop X)) W ≫ colimMap (J.diagramNatTrans η (unop X))) ≫ (J.plusCompIso F Q).hom.app X =
    ((J.diagramCompIso F P (unop X)).hom.app W ≫
        Multiequalizer.lift ((unop W).index (Q ⋙ F)) ((J.diagram (P ⋙ F) (unop X)).obj W)
          (fun x => Multiequalizer.ι ((unop W).index (P ⋙ F)) x ≫ F.map (η.app (op x.Y))) ⋯) ≫
      colimit.ι (J.diagram (Q ⋙ F) (unop X)) W

Tactic:
dsimp [colimMap, IsColimit.map]

Result:
case w.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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ F.map
        (colimit.ι (J.diagram P (unop X)) W ≫
          colimit.desc (J.diagram P (unop X))
            ((Cocones.precompose (J.diagramNatTrans η (unop X))).obj (colimit.cocone (J.diagram Q (unop X))))) ≫
      (J.plusCompIso F Q).hom.app X =
    ((J.diagramCompIso F P (unop X)).hom.app W ≫
        Multiequalizer.lift ((unop W).index (Q ⋙ F)) (multiequalizer ((unop W).index (P ⋙ F)))
          (fun x => Multiequalizer.ι ((unop W).index (P ⋙ F)) x ≫ F.map (η.app (op x.Y))) ⋯) ≫
      colimit.ι (J.diagram (Q ⋙ F) (unop X)) W

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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
a : (unop W).shape.L
⊢ (F.map
          (Multiequalizer.lift ((unop W).index Q) (multiequalizer ((unop W).index P))
            (fun x => Multiequalizer.ι ((unop W).index P) x ≫ η.app (op x.Y)) ⋯) ≫
        (J.diagramCompIso F Q (unop X)).hom.app W) ≫
      Multiequalizer.ι ((unop W).index (Q ⋙ F)) a =
    ((J.diagramCompIso F P (unop X)).hom.app W ≫
        Multiequalizer.lift ((unop W).index (Q ⋙ F)) (multiequalizer ((unop W).index (P ⋙ F)))
          (fun x => Multiequalizer.ι ((unop W).index (P ⋙ F)) x ≫ F.map (η.app (op x.Y))) ⋯) ≫
      Multiequalizer.ι ((unop W).index (Q ⋙ F)) a

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

Result:
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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
a : (unop W).shape.L
⊢ F.map
        (Multiequalizer.lift ((unop W).index Q) (multiequalizer ((unop W).index P))
          (fun x => Multiequalizer.ι ((unop W).index P) x ≫ η.app (op x.Y)) ⋯) ≫
      F.map (Multiequalizer.ι ((unop W).index Q) a) =
    F.map (Multiequalizer.ι ((unop W).index P) a) ≫ F.map (η.app (op a.Y))

Proof state:
case w.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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ F.map (colimit.ι (J.diagram P (unop X)) W) ≫
      F.map (colimMap (J.diagramNatTrans η (unop X))) ≫ (J.plusCompIso F Q).hom.app X =
    F.map (colimit.ι (J.diagram P (unop X)) W) ≫
      (J.plusCompIso F P).hom.app X ≫ colimMap (J.diagramNatTrans (whiskerRight η F) (unop X))

Tactic:
simp only [ι_colimMap, whiskerRight_app, ι_plusCompIso_hom_assoc,
  GrothendieckTopology.diagramNatTrans_app]

Result:
case w.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 : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E
inst✝³ : ∀ (X : C) (W : J.Cover X) (P : Cᵒᵖ ⥤ D), PreservesLimit (W.index P).multicospan F
inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E
inst✝ : ∀ (X : C), PreservesColimitsOfShape (J.Cover X)ᵒᵖ F
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
X : Cᵒᵖ
W : (J.Cover (unop X))ᵒᵖ
⊢ F.map (colimit.ι (J.diagram P (unop X)) W) ≫
      F.map (colimMap (J.diagramNatTrans η (unop X))) ≫ (J.plusCompIso F Q).hom.app X =
    (J.diagramCompIso F P (unop X)).hom.app W ≫
      Multiequalizer.lift ((unop W).index (Q ⋙ F)) ((J.diagram (P ⋙ F) (unop X)).obj W)
          (fun x => Multiequalizer.ι ((unop W).index (P ⋙ F)) x ≫ F.map (η.app (op x.Y))) ⋯ ≫
        colimit.ι (J.diagram (Q ⋙ F) (unop X)) W