Let $\mathcal{C}$ be a category. A family of morphisms in $\mathcal{C}$ is said be parallel if they belong to $\operatorname{Hom}(A,B)$ for some objects $A,B$ in $\mathcal{C}$

Let $f,g$ be a pair of parallel morphisms in $\operatorname{Hom}(A,B)$ A morphism $d\colon X\to A$ is said to equalize $f$ and $g$ if $fd=gd$ In other words, the following diagrams are equal: $$\xymatrix@1{X\ar[r]^d&A\ar[r]^f&B}=\xymatrix@1{X\ar[r]^d&A\ar[r]^g&B}$$

An equalizer of $f$ and $g$ is a morphism $d$ from an object $X \in \mathcal{C}$ to $A$ such that

1. $d$ equalizes $f$ and $g$
2. $d$ is universal among all morphisms that equalize $f$ and $g$ Specifically, if $e$ is a morphism from an object $Y\in\mathcal{C}$ to $A$ such that $e$ equalizes $f$ and $g$ then there exists a unique morphism $h:Y\to X$ and a commutative diagram: $$\xymatrix@1{Y \ar[d]_h \ar[dr]^e \\ X \ar[r]_d & A}$$

1. $\xymatrix@1{A\ar[r]^f&B\ar[r]^c&Z}=\xymatrix@1{A\ar[r]^g&B\ar[r]^c&Z}$ Such a morphism is said to coequalize $f$ and $g$
2. $c$ is universal among all morphisms that coequalizes $f$ and $g$ This means that given a morphism $r$ from $B$ to an object $Y\in\mathcal{C}$ there exists a unique morphism $r\in\operatorname{Hom}(Z,Y)$ so the following diagram commutes: $$\xymatrix@1{B \ar[dr]_e \ar[r]^c & Z \ar[d]^r \\ & Y}$$

Remarks

• An equalizer is a monomorphism (but not the other way around, a monomorphism that is also an equalizer is called a regular monomorphism). A coequalizer is an epimorphism (and conversely, an epimorphism that is also a coequalizer is called a regular epimorphism). This follows directly from the above definitions and definitions of monomorphisms and epimorphisms.
• If $X\to A$ is an equalizer of $f,g\colon A\to B$ then $[X\to A]$ is a subobject of $A$ Furthermore, by the universality of the equalizer, it is the ``largest'' such subobject. Similarly, If $B\to Z$ is a coequalizer of $f,g$ then $[B\to Z]$ is the ``largest" quotient object of $B$
• From the above discussion, we can safely say the equalizer of $f$ and $g$ and the coequalizer of $f$ and $g$
• The equalizer of a morphism $f:A\to B$ and itself is the identity morphism $1_A$ on $A$
• A category is said to have equalizers if every pair of parallel morphisms has an equalizer.

One can also define an equalizer of an arbitrary set of morphisms with a common domain and a common codomain: if $\lbrace f_i:A\to B\mid i\in I\rbrace$ is a set of morphisms from $A$ to $B$ indexed by a set $I$ then an equalizer of the $f_i$ s is a morphism $d$ from an object $X$ to $A$ such that $d$ equalizes every pair of morphisms $f_i$ and $f_j$ and that $d$ is universal among all morphisms with such a property.

Remark. An equalizer (coequalizer) is also known as a difference kernel (difference cokernel). This name is justifiably given as we recognize that a kernel of a morphism $f$ is, in a way, the ``difference" between $f$ and $o$ the zero morphism.