map_id — Mathlib

English

MapDesc is compatible with composition: mapDesc f g (i ∘ i') equals mapDesc f g i followed by mapDesc _ _ i' and a coherence isom.

Русский

MapDesc совместим с композициями: mapDesc f g (i ∘ i') равен mapDesc f g i затем mapDesc _ _ i' и когерентное изоморфие.

LaTeX

$$$\mathrm{pullback.mapDesc} f g (i \circ i') = \mathrm{pullback.mapDesc} f g i \, \circ \, \mathrm{pullback.mapDesc} _ _ i' \circ (\mathrm{pullback.congrHom} (\mathrm{Category.assoc} \_ \_ \_) (\mathrm{Category.assoc} \_ \_ \_)).hom$$$

Lean4

@[simp]
theorem map_id {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} [HasPullback f g] :
    pullback.map f g f g (𝟙 _) (𝟙 _) (𝟙 _) (by simp) (by simp) = 𝟙 _ := by ext <;> simp

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