Equivalence of categories is reflexive, symmetric, and transitive

The identity functor is its own quasi-inverse and the identity natural transformations supply an equivalence $\mathcal C\simeq\mathcal C$, proving reflexivity.

If $(F,G,\eta,\varepsilon)$ gives $\mathcal C\simeq\mathcal D$, then $(G,F,\varepsilon^{-1},\eta^{-1})$ gives $\mathcal D\simeq\mathcal C$, proving symmetry.

If $(F,G)$ gives $\mathcal C\simeq\mathcal D$ and $(H,K)$ gives $\mathcal D\simeq\mathcal E$, then $HF$ and $GK$ are quasi-inverses; whiskering and vertically composing the two units gives $1_{\mathcal C}\Rightarrow GKHF$, and doing the same with the counits gives $HFGK\Rightarrow1_{\mathcal E}$, so [L2] proves transitivity.