Equivalence, quasi-inverse, and adjoint equivalence of categories

An equivalence of categories from $\mathcal C$ to $\mathcal D$ consists of functors $F:\mathcal C\to\mathcal D$ and $G:\mathcal D\to\mathcal C$, called quasi-inverses, together with natural isomorphisms

$\eta:1_{\mathcal C}\Rightarrow GF,\qquad \varepsilon:FG\Rightarrow1_{\mathcal D}.$

The categories are then called equivalent. The notions of functor and natural isomorphism are those of Covariant functor, identity functor, composite functor, and contravariant functor and Natural isomorphism.

An adjoint equivalence is such data satisfying the triangle identities

$G\varepsilon\circ\eta G=1_G,\qquad \varepsilon F\circ F\eta=1_F.$

Here $G\varepsilon$, $\eta G$, $\varepsilon F$, and $F\eta$ are the whiskerings of Whiskering and horizontal composition of natural transformations. No triangle identity is required of a bare equivalence.