A strong monomorphism has the right lifting property with respect to every epimorphism. In every commutative square with an epimorphism on the left and the strong monomorphism on the right, there is a unique diagonal filler; uniqueness follows from monicity.
A regular monomorphism is a morphism that occurs as the equalizer of some parallel pair. Every regular monomorphism is a strong monomorphism.
Whenever the relevant pullback exists, the intersection of two strong subobjects of an object is strong. More generally, an arbitrary small intersection of strong subobjects is strong when it exists: lift into each subobject and use their common composite into the ambient object to obtain a map into the limit.
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