-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon.{v₁, u₁} C
M : Bimod W X
N N' : Bimod X Y
f : N ⟶ N'
P : Bimod Y Z
dsimp :
  Eq (α :=
    {
        X :=
          TensorBimod.X
            { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod W Y }
            P,
        actLeft :=
          TensorBimod.actLeft
            { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod W Y }
            P,
        one_actLeft := ⋯, left_assoc := ⋯,
        actRight :=
          TensorBimod.actRight
            { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod W Y }
            P,
        actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ : Bimod W Z } ⟶
      {
        X :=
          TensorBimod.X
            { X := TensorBimod.X M N', actLeft := TensorBimod.actLeft M N', one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight M N', actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
              Bimod W Y }
            P,
        actLeft :=
          TensorBimod.actLeft
            { X := TensorBimod.X M N', actLeft := TensorBimod.actLeft M N', one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight M N', actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }
            P,
        one_actLeft := ⋯, left_assoc := ⋯,
        actRight :=
          TensorBimod.actRight
            { X := TensorBimod.X M N', actLeft := TensorBimod.actLeft M N', one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight M N', actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }
            P,
        actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ })
    (whiskerRight (M.whiskerLeft f) P)
    (Iso.hom
        (isoOfIso
          { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ :
            TensorBimod.X
                { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                  Bimod W Y }
                P ≅
              TensorBimod.X M
                { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                  Bimod X Z } }
          ⋯ ⋯) ≫
      M.whiskerLeft (whiskerRight f P) ≫
        Iso.inv
          (isoOfIso
            { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
            ⋯))
⊢ Eq (α := (M.tensorBimod N).tensorBimod P ⟶ (M.tensorBimod N').tensorBimod P) (whiskerRight (M.whiskerLeft f) P)
    (Iso.hom (M.associatorBimod N P) ≫ M.whiskerLeft (whiskerRight f P) ≫ Iso.inv (M.associatorBimod N' P))

Tactic:
dsimp [tensorHom, tensorBimod, associatorBimod]

Premises used: Bimod.tensorBimod, Bimod.associatorBimod, CategoryTheory.MonoidalCategoryStruct.tensorHom