Definition 3.1 (Adjunction, D. Kan 1958). Let $C$ and $D$ be categories. An adjunction between $C$ and $D$ consists of functors $F:C\to D$ and $G:D\to C$ together with, for each $A\in \mathrm{ob}C$ and $B\in \mathrm{ob}D$, a bijection between morphisms $FA\to B$ in $D$ and morphisms $A\to GB$ in $C$, which is natural in $A$ and $B$. (If $C$ and $D$ are locally small, this means that $D(F\bullet ,\bullet )$ and $C(\bullet ,G\bullet )$ are naturally isomorphic functors $C^{op}\times D\to Set$.)

We say $F$ is left adjoint to $G$, or $G$ is right adjoint to $F$, and we write $(F⊣G)$.