A category consists of\n1. a collection of objects $X, Y, Z, \ldots$\n2. a collection of morphisms $f, g, h, \ldots $\nso that:\n1. Each morphism has specified domain and codomain objects; the notation $f \colon X \to Y$ signifies that $f$ is a morphism with domain $X$ and codomain $Y$.\n2. Each object has a designated identity morphism $1_X \colon X \to X$.\n3. For any pair of morphisms $f,g$ with the codomain of $f$ equal to the domain of $g$, there exists a specified composite morphism $gf$ whose domain is equal to the domain of $f$ and whose codomain is equal to the codomain of $g$, i.e.,:\n\(f \colon X \to Y,\quad g \colon Y \to Z \qquad \rightsquigarrow\qquad gf \colon X \to Z.\)\nThis data is subject to the following two axioms:\n1. For any $f \colon X \to Y$, the composites $1_Y f$ and $f 1_X$ are both equal to $f$.\n2. For any composable triple of morphisms $f,g,h$, the composites $h(gf)$ and $(hg)f$ are equal and henceforth denoted by $hgf$.\n\(f \colon X \to Y,\quad g \colon Y \to Z,\quad h \colon Z \to W \qquad \rightsquigarrow\qquad hgf \colon X \to W.\)\nThat is, the composition law is associative and unital with the identity morphisms serving as two-sided identities.