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src/Categories/Bicategory/Construction/Bimodules/Tensorproduct/Unitor.agda

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@@ -134,7 +134,7 @@ module Left-Unitor where
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≈ (λ⇒-⊗ ∘ᵥ actionˡ (Id-Bimod ⊗₀ B)) ∘ᵥ arr (CoeqBimods Id-Bimod B) ◁ T M₁
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linearˡ-∘arr = begin
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(actionˡ B ∘ᵥ λ⇒-⊗ ◁ T M₁) ∘ᵥ arr (CoeqBimods Id-Bimod B) ◁ T M₁ ≈⟨ glue◽▹ (⟺ (assoc B)) (◁-resp-tri triangle) ⟩
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actionʳ B ∘ᵥ actionˡ-∘ ≈⟨ ⟺ (glue◃◽ triangle (⟺ actionˡSq-⊗)) ⟩
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actionʳ B ∘ᵥ actionˡ-∘ ≈⟨ ⟺ (glue◃◽ triangle (⟺ actionˡSq-⊗)) ⟩
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(λ⇒-⊗ ∘ᵥ actionˡ (Id-Bimod ⊗₀ B)) ∘ᵥ arr (CoeqBimods Id-Bimod B) ◁ T M₁ ∎
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where
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open hom.HomReasoning
@@ -278,7 +278,7 @@ module Right-Unitor where
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≈ (ρ⇒-⊗ ∘ᵥ actionˡ (B ⊗₀ Id-Bimod)) ∘ᵥ arr (CoeqBimods B Id-Bimod) ◁ T M₁
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linearˡ-∘arr = begin
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(actionˡ B ∘ᵥ ρ⇒-⊗ ◁ T M₁) ∘ᵥ arr (CoeqBimods B Id-Bimod) ◁ T M₁ ≈⟨ glue◽▹ (⟺ (assoc-actionˡ B)) (◁-resp-tri triangle) ⟩
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actionˡ B ∘ᵥ actionˡ-∘ ≈⟨ ⟺ (glue◃◽ triangle (⟺ actionˡSq-⊗)) ⟩
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actionˡ B ∘ᵥ actionˡ-∘ ≈⟨ ⟺ (glue◃◽ triangle (⟺ actionˡSq-⊗)) ⟩
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(ρ⇒-⊗ ∘ᵥ actionˡ (B ⊗₀ Id-Bimod)) ∘ᵥ arr (CoeqBimods B Id-Bimod) ◁ T M₁ ∎
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where
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open hom.HomReasoning
@@ -303,7 +303,7 @@ module Right-Unitor where
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≈ (ρ⇒-⊗ ∘ᵥ actionʳ (B ⊗₀ Id-Bimod)) ∘ᵥ T M₂ ▷ arr (CoeqBimods B Id-Bimod)
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linearʳ-∘arr = begin
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(actionʳ B ∘ᵥ T M₂ ▷ ρ⇒-⊗) ∘ᵥ T M₂ ▷ arr (CoeqBimods B Id-Bimod) ≈⟨ glue◽▹ (⟺ (sym-assoc B)) (▷-resp-tri triangle) ⟩
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actionˡ B ∘ᵥ actionʳ-∘ ≈⟨ ⟺ (glue◃◽ triangle (⟺ actionʳSq-⊗)) ⟩
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actionˡ B ∘ᵥ actionʳ-∘ ≈⟨ ⟺ (glue◃◽ triangle (⟺ actionʳSq-⊗)) ⟩
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(ρ⇒-⊗ ∘ᵥ actionʳ (B ⊗₀ Id-Bimod)) ∘ᵥ T M₂ ▷ arr (CoeqBimods B Id-Bimod) ∎
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where
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open hom.HomReasoning

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