factorThruKernelSubobject_comp_kernelSubobjectIso

English

For a commuting square sq between arrows f and f', the kernelSubobjectMap sq composed with (kernelSubobject f').arrow equals (kernelSubobject f).arrow composed with sq.left.

Русский

Для квадратной последовательности между стрелами f и f' карта kernelSubobject sq переводит стрелу ядро-подобъекта в соответствующую стрелу f'.

LaTeX

$$$$ kernelSubobjectMap\\ sq \\; \\gg (kernelSubobject f').arrow = (kernelSubobject f).arrow \\; \\gg sq.left $$$$

Lean4

@[simp]
theorem factorThruKernelSubobject_comp_kernelSubobjectIso {W : C} (h : W ⟶ X) (w : h ≫ f = 0) :
    factorThruKernelSubobject f h w ≫ (kernelSubobjectIso f).hom = kernel.lift f h w :=
  (cancel_mono (kernel.ι f)).1 <| by simp

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