CatDat

Implication Details

Claim: If a morphism is a regular monomorphism, then it is a strict monomorphism.

Proof: Let $m : A \to B$ be the equalizer of $g,h : B \rightrightarrows C$. In particular, $m$ is a monomorphism. Let $t : T \to B$ be a monomorphism which equalizes all pairs that are equalized by $m$. In particular, $t$ equalizes $g,h$, i.e. $g \circ t = h \circ t$. By definition of an equalizer, this means that $t$ factors through $m$.