iso_hom_app — Mathlib

English

Inverse components of the shifted zero functor for φ0 factor through the pullback iso, giving a simplified expression in the φ0-shifted context.

Русский

Обратные компоненты нулевого сдвига по φ0 проходят через pullback iso, упрощая выражение в контексте φ0-сдвига.

LaTeX

$$$ (shiftFunctorZero\; (PullbackShift C φ) A)^{-1}_{X} = (shiftFunctorZero' C (φ 0) (by rw[map_zero])).inv.app X \circ (pullbackShiftIso C φ 0 (φ 0) rfl)^{-1}_{X} $$$

Lean4

@[simp]
theorem iso_hom_app (a : A) (X : C) :
    (iso F r hF a).hom.app ((functor r).obj X) =
      (lift r F hF).map (((functor r).commShiftIso a).inv.app X) ≫ (F.commShiftIso a).hom.app X :=
  by
  dsimp only [iso, natIsoLift]
  rw [natTransLift_app]
  dsimp
  erw [comp_id, id_comp, id_comp, id_comp, Functor.map_id, comp_id]

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