CategoryTheory.effectiveEpiOfKernelPair

theorem CategoryTheory.effectiveEpiOfKernelPair : ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] {B X : C} (f : X ⟶ B) [inst_1 : CategoryTheory.Limits.HasPullback f f], CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.Cofork.ofπ f ⋯) → CategoryTheory.EffectiveEpi f

This theorem in the Category Theory states that for any category `C` and any two objects `B` and `X` in `C`, if there exists a pullback for the morphism `f` from `X` to `B` with itself (which forms a so-called kernel pair), and if this morphism `f` also serves as a coequalizer for its kernel pair, then `f` is an effective epimorphism. In simpler terms, it says that if a morphism balances the property of being a pullback of itself and serving as a coequalizer for its kernel pair, it effectively 'spreads out' information from `X` to `B` without any loss, hence being an effective epimorphism.

More concisely: If a morphism in a category is both the pullback of itself and the coequalizer of its kernel pair, then it is an effective epimorphism.