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[parent] proof that an equalizer is a monomorphism (Proof)

Suppose that $f$ is the equalizer of $g$ and $h$ (refer to the commutative diagram below). Let $a,b$ be morphisms such that

\begin{displaymath}fa=fb=d.\end{displaymath}

We must show that $a=b$. Since $d$ equalizes $g$ and $h$, by definition of equalizer $c$ is the unique morphsim such that $d=fc$. Hence, both $a$ and $b$ are equal to $c$, and hence equal to each other.
Figure: An equalizer is a monomorphism.
\includegraphics[width=8cm]{equalizermono-fig}



"proof that an equalizer is a monomorphism" is owned by rmilson.
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Cross-references: morphisms, commutative diagram, equalizer
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This is version 3 of proof that an equalizer is a monomorphism, born on 2006-02-16, modified 2006-06-06.
Object id is 7627, canonical name is ProofThatAnEqualizerIsAMonomorphism.
Accessed 738 times total.

Classification:
AMS MSC: 18A20 (Category theory; homological algebra :: General theory of categories and functors :: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms)
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