= Extremal monomorphism
A <monomorphism> $m$ is extremal if every factorization $m=he$ with $e$ an <epimorphism> forces $e$ to be an <isomorphism>. A <strong monomorphism> has this property by its lifting property. In a <complete category> that is a <well-powered category>, intersecting all strong <subobjects> containing a given morphism yields an epimorphism–strong-monomorphism factorization; applying it to an extremal monomorphism proves that the extremal monomorphism is strong. The factorization definition and lifting definition should be kept distinct until the needed ambient hypotheses are supplied.
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