A category $\mathscr{A}$ consists of: 1. a collection $\ob(\mathscr{A})$ of objects;\n2. for each $A, B \in \ob(\mathscr{A})$, a collection $\mathscr{A}(A, B)$ of maps or arrows or morphisms from $A$ to $B$;\n3. for each $A, B, C \in \ob(\mathscr{A})$, a function \(\begin{array}{ccc} \mathscr{A}(B, C) \times \mathscr{A}(A, B) & \to & \mathscr{A}(A, C) \\ (g, f) & \mapsto & g \circ f, \end{array}\) called composition;\n4. for each $A \in \ob(\mathscr{A})$, an element $1_A$ of $\mathscr{A}(A, A)$, called the identity on $A$, satisfying the following axioms: 1. associativity: for each $f \in \mathscr{A}(A, B)$, $g \in \mathscr{A}(B, C)$ and $h \in \mathscr{A}(C, D)$, we have $(h \circ g) \circ f = h \circ (g \circ f)$;\n2. identity laws: for each $f \in \mathscr{A}(A, B)$, we have $f \circ 1_A = f = 1_B \circ f$.