proposition: Lemma are_inverses_from_Kernel_to_Kernel {y z : C} {g : y --> z} (K K': Kernel g) : is_inverse_in_precat (from_Kernel_to_Kernel K K') (from_Kernel_to_Kernel K' K).
proof: Proof. split. - apply pathsinv0. use KernelEndo_is_identity. rewrite <- assoc. unfold from_Kernel_to_Kernel. rewrite KernelCommutes. rewrite KernelCommutes. apply idpath. - apply pathsinv0. use KernelEndo_is_identity. rewrite <- assoc. unfold from_Kernel_to_Kernel. rewrite KernelCommutes. rewrite KernelCommutes. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Zero