An equivalence between categories $\mathscr{A}$ and $\mathscr{B}$ consists of a pair~ of functors together with natural isomorphisms \(\eta{\colon}\linebreak[0] 1_\mathscr{A} \to G \circ F, \qquad \epsln{\colon}\linebreak[0] F \circ G \to 1_\mathscr{B}.\) If there exists an equivalence between $\mathscr{A}$ and $\mathscr{B}$, we say that $\mathscr{A}$ and $\mathscr{B}$ are equivalent, and write $\mathscr{A} \simeq \mathscr{B}$. We also say that the functors $F$ and $G$ are equivalences.