An equalizer is a limit
over a parallel pair i.e. of the diagram of the shape
(See also fork diagram).
This means that for $f : x \to y$ and $g : x \to y$ two parallel morphisms in a category $C$, their equalizer is, if it exists
an object $eq(f,g) \in C$;
a morphism $eq(f,g) \to x$
such that
The dual concept is that of coequalizer.
In type theory the equalizer
is given by the dependent sum over the dependent equality type
In $C =$ Set the equalizer of two functions of sets is the subset of elements of $c$ on which both functions coincide.
For $C$ a category with zero object the equalizer of a morphism $f : c \to d$ with the corresponding zero morphism is the kernel of $f$.
For $S \stackrel{\overset{g}{\longrightarrow}}{\underset{f}{\longrightarrow}} T$ the given diagram, form the pullback along the diagonal morphism of $T$:
One checks that the horizontal morphism $eq(f,g) \to S$ equalizes $f$ and $g$ and that it does so universally.
If a category has equalizers and (finite) products, then it has (finite) limits.
For the finite case, we may say equivalently:
If a category has equalizers, binary products and a terminal object, then it has finite limits.
(Eckmann and Hilton EH, Proposition 1.3.) Let $e: E \rightarrow X$ be an arrow in a category $\mathcal{C}$ which is an equaliser of a pair of arrows of $\mathcal{C}$. Then $e$ is a monomorphism.
If $g,h : A \rightarrow E$ are arrows of $\mathcal{C}$ such that $e \circ g = e \circ h$, then it follows immediately from the uniqueness part of the universal property of an equaliser that $g = h$.
Equalizers were defined in the paper
for any finite collection of parallel morphisms. The paper refers to them as left equalizers, whereas coequalizers are referred to as right equalizers.
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