-- 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)
X Y Z : Mon.{v₁, u₁} C
M : Bimod X Y
N : Bimod Y Z
h :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) N.X ⟶ TensorBimod.X M N)
    (Iso.hom (α_ M.X Y.X N.X) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) M.X (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
        (M.X ◁ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft))
        (M.X ◁ coequalizer.desc N.actLeft ⋯ ≫
          colimit.ι (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))
            WalkingParallelPair.one))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ Y.X) (Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ)) N.X)
      (coequalizer.π (M.actRight ▷ Y.X) (Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ) ▷ N.X)
      (coequalizer.desc M.actRight ⋯ ▷ N.X ≫
        colimit.ι (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))
          WalkingParallelPair.one))
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X) N.X ⟶ TensorBimod.X M N)
    (Iso.hom (α_ M.X Y.X N.X) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) M.X (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
        (M.X ◁ coequalizer.π (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) Y.X N.X) ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst) M.X (coequalizer (μ ▷ N.X) (Iso.hom (α_ Y.X Y.X N.X) ≫ Y.X ◁ N.actLeft)))
        (M.X ◁ LeftUnitorBimod.hom N ≫
            colimit.ι (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))
              WalkingParallelPair.one :
          (parallelPair (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X)
                    (TensorBimod.X
                      { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                        right_assoc := ⋯, middle_assoc := ⋯ : Bimod Y Y }
                      N))
                  (M.actRight ▷
                    TensorBimod.X
                      { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                        right_assoc := ⋯, middle_assoc := ⋯ : Bimod Y Y }
                      N)
                  (Iso.hom
                      (α_ M.X Y.X
                        (TensorBimod.X
                          { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                            right_assoc := ⋯, middle_assoc := ⋯ : Bimod Y Y }
                          N)) ≫
                    M.X ◁
                      TensorBimod.actLeft
                        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯,
                          right_assoc := ⋯, middle_assoc := ⋯ : Bimod Y Y }
                        N)).obj
              WalkingParallelPair.one ⟶
            colimit (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ Y.X) (Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ)) N.X)
      (coequalizer.π (M.actRight ▷ Y.X) (Iso.hom (α_ M.X Y.X Y.X) ≫ M.X ◁ μ) ▷ N.X)
      (coequalizer.desc M.actRight ⋯ ▷ N.X ≫
        colimit.ι (parallelPair (M.actRight ▷ N.X) (Iso.hom (α_ M.X Y.X N.X) ≫ M.X ◁ N.actLeft))
          WalkingParallelPair.one))

Tactic:
dsimp [LeftUnitorBimod.hom]

Premises used: Bimod.LeftUnitorBimod.hom