English
A compatibility under isomorphisms of B with Z yields a transformation of imageToKernel along Iso f i.hom and i.inv g by a standard iso composition.
Русский
Совместимость через изоморфизмы: imageToKernel через изоморфизм B ≅ Z трансформируется аналогично.
LaTeX
$$[HasEqualizers V] [HasImages V] → imageToKernel (f ≫ i.hom) (i.inv ≫ g) w = (imageSubobjectCompIso _ _).hom ≫ imageToKernel f g (by simpa using w) ≫ (kernelSubobjectIsoComp i.inv g).inv$$
Lean4
@[simp]
theorem imageToKernel_epi_comp {Z : V} (h : Z ⟶ A) [Epi h] (w) :
imageToKernel (h ≫ f) g w =
Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫
imageToKernel f g ((cancel_epi h).mp (by simpa using w : h ≫ f ≫ g = h ≫ 0)) :=
by
ext
simp