liftCommShift_compatibility — Mathlib

English

Hom component of the φ0-shifted zero functor factors as the pullback shift iso followed by the φ0-shifted base hom.

Русский

Гом-компонента φ0-сдвиговка нулевого оператора раскладывается через pullbackShift iso и гом базового φ0-сдвига.

LaTeX

$$$ (\mathrm{shiftFunctorZero} (\mathrm{PullbackShift} \ C \ varphi) A)^{hom}_{X} = (\mathrm{pullbackShiftIso} C\ varphi 0 (\varphi 0) rfl)^{hom}_{X} \circ (\mathrm{shiftFunctorZero'} C (\varphi 0) (by rw[map_zero])).hom_{X} $$$

Lean4

instance liftCommShift_compatibility : NatTrans.CommShift (Quotient.lift.isLift r F hF).hom A where
  shift_comm
    a := by
    ext X
    dsimp
    erw [Functor.map_id, id_comp, comp_id]
    rw [Functor.commShiftIso_comp_hom_app]
    erw [LiftCommShift.iso_hom_app]
    rw [← Functor.map_comp_assoc, Iso.hom_inv_id_app, Functor.map_id, id_comp]

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