A category consists of a collection of objects with morphisms between them. A morphism goes from one object, say $X$ , to another, say $Y$ , and is drawn as an arrow from $X$ to $Y$ . Note that $X$ may equal $Y$ (in which case $f$ is referred to as an endomorphism). The object $X$ is called the source or domain of $f$ and $Y$ is called the target or codomain of $f$ , though note that $f$ itself need not be a function and $X$ and $Y$ need not be sets. This is written as $f : X \to Y$ .

These morphisms must satisfy three conditions:

Composition: For any two morphisms $f : X \to Y$ and $g : Y \to Z$ , there exists a morphism $X \to Z$ , written as $g \circ f$ or simply $g f$ .

Associativity: For any morphisms $f : X \to Y$ , $g : Y \to Z$ and $h : Z \to W$ composition is associative, i.e., $h (g f) = (h g) f$ .

Identity: For any object $X$ , there is a (unique) morphism, $1_{X} : X \to X$ which, when composed with another morphism, leaves it unchanged. I.e., given $f : W \to X$ and $g : X \to Y$ it holds that: $1_{X} f = f$ and $g 1_{X} = g$ .

Category (mathematics)