English
If two subobjects mk f and mk g of B satisfy mk f ≤ mk g, then there exists a morphism A1 → A2 witnessing the inclusion of the subobject represented by f into the one represented by g.
Русский
Если mk f и mk g — подобъект B, удовлетворяющий mk f ≤ mk g, то существует мороморфизм A1 → A2, который свидетельствует о включении подобъекта, задаваемого f, в подобъект, задаваемый g.
LaTeX
$$$\exists \varphi : A_1 \to A_2 \,\text{such that}\\ \varphi \;\;\;g = f \,\;\text{and}\; \, \text{the witness } \varphi \text{ is } (\text{underlyingIso } f)^{-1} \;\rhd\; \operatorname{ofLE}(\mk f)(\mk g)h \;\rhd\; (\text{underlyingIso } g)\!\text{(hom)}$$$
Lean4
/-- An inequality of subobjects is witnessed by some morphism between the corresponding objects. -/
def ofMkLEMk {B A₁ A₂ : C} (f : A₁ ⟶ B) (g : A₂ ⟶ B) [Mono f] [Mono g] (h : mk f ≤ mk g) : A₁ ⟶ A₂ :=
(underlyingIso f).inv ≫ ofLE (mk f) (mk g) h ≫ (underlyingIso g).hom
