Let $C$ and $D$ be two categories with functors $F\colon C \to D$ and $G\colon D \to C$ . The functors $F$ and $G$ are an equivalence of categories if there are natural isomorphisms $FG \cong \id_D$ and $GF \cong \id_C$ .

Note, $F$ is left adjoint to $G$ , and $G$ is right adjoint to $F$ as$$ \hom_D(F(c),d) \stackrel{G}{\longrightarrow} \hom_C(GF(c),G(d)) \longleftrightarrow \hom_C(c,G(d)).$$ And, $F$ is right adjoint to $G$ , and $G$ is left adjoint to $F$ as$$ \hom_C(G(d),c) \stackrel{F}{\longrightarrow} \hom_D(FG(d),F(c)) \longleftrightarrow \hom_D(d,F(c)).$$

In practical terms, two categories are equivalent if there is a fully faithful functor $F\colon C \to D$ , such that every object $d \in D$ is isomorphic to an object $F(c)$ , for some $c \in C$ .