Category
A category \(\mathscr{C}\) is a collection of objects, denoted \(\text{obj}{\mathscr{C}}\) together with a collection of classes of morphisms between pairs of objects, each of which is denoted \(\text{Mor}{(a,b)}\) for any pair \(a,b \in \text{obj}{\mathscr{C}}\), such that the following data is given:
(1). For any objects \(a,b,c \in \text{obj}{\mathscr{C}}\), there exist composition maps
\[ \circ:\text{Mor}{(b,c)} \times \text{Mor}{(a,b)} \rightarrow \text{Mor}{(a,c)} \]associating to each pair of morphisms \((g,f)\) a morphism \(g \circ f\) called the composition of the morphism \(g\) with the morphism \(f\).
(2). For any objects \(a,b,c,d \in \text{obj}{\mathscr{C}}\), and for any morphisms \(f \in \text{Mor{(a,b)}}\), \(g \in \text{Mor{(b,c)}}\), and \(h \in \text{Mor{(c,d)}}\), the composition maps between morphisms associate wherever they exist; that is:
\[ h \circ (g \circ f)=(h \circ g) \circ f \]whenever either \(h \circ (g \circ f)\) or \((h \circ g) \circ f\) exists.
(3). For any object \(a \in \text{obj}{\mathscr{C}}\), there is a morphism \(id_a \in \text{Mor}{(a,a)}\), called the identity morphism on \(a\), such that for any objects \(b,c \in \text{obj}{\mathscr{C}}\), and for any morphisms \(f \in \text{Mor}{(a,b)}\) and \(g \in \text{Mor}{(c,a)}\), the following holds:
\[ f \circ id_a=f \text{ and } id_a \circ g=g \]Given objects \(a,b \in \text{obj}{\mathscr{C}}\), we denote a morphism \(f \in \text{Mor}{(a,b)}\) by \(f:a \rightarrow b\).
(sidenote: Here we take the terms _object_ and _morphism_ to be undefined. )