Definition:Equalizer

Definition
Let $\mathbf C$ be a metacategory.

Let $f, g: C \to D$ be morphisms with common domain and codomain.

An equalizer for $f$ and $g$ is a morphism $e: E \to C$ such that:


 * $f \circ e = g \circ e$

and subject to the following UMP:


 * For any $a: A \to C$ such that $f \circ a = g \circ a$, there is a unique $u: A \to E$ such that:


 * $\begin{xy}

<-4em,0em>*+{E} = "E", <0em,0em>*+{C}  = "C", <4em,0em>*+{D}  = "D", <-4em,-4em>*+{A} = "A",

"E";"C" **@{-} ?>*@{>} ?*!/_.8em/{e}, "C"+/^.3em/+/r.5em/;"D"+/^.3em/+/l.5em/ **@{-} ?>*@{>} ?*!/_.8em/{f}, "C"+/_.3em/+/r.5em/;"D"+/_.3em/+/l.5em/ **@{-} ?>*@{>} ?*!/^.8em/{g}, "A";"E" **@{.} ?>*@{>} ?*!/_.8em/{u}, "A";"C" **@{-} ?>*@{>} ?*!/^.8em/{a}, \end{xy}$


 * is a commutative diagram. I.e., $a = e \circ u$.

Also see

 * Coequalizer, the dual notion