18 lines
737 B
Text
18 lines
737 B
Text
import logic algebra.category.basic
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open eq eq.ops category functor natural_transformation
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variables {obC obD : Type} {C : category obC} {D : category obD} {F G H : C ⇒ D}
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protected definition compose2 (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
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natural_transformation.mk
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(λ a, η a ∘ θ a)
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(λ a b f, calc
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H f ∘ (η a ∘ θ a) = (H f ∘ η a) ∘ θ a : assoc
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... = (η b ∘ G f) ∘ θ a : naturality η f
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... = η b ∘ (G f ∘ θ a) : assoc
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... = η b ∘ (θ b ∘ F f) : naturality θ f
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... = (η b ∘ θ b) ∘ F f : assoc)
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theorem tst (a b c : num) (H₁ : ∀ x, b = x) (H₂ : c = b) : a = c :=
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calc a = b : H₁
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... = c : H₂
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