Infinity Category Adjunction

An adjuction between infinity categories consists of:

• A pair of infinity categories A,B
• A pair of functors $f:B\Rightarrow A$, $g:A\Rightarrow B$
• A pair of natural transformations $\eta :i{d}_B\Rrightarrow gf$, $ϵ:fg\Rrightarrow i{d}_A$ Such that, Write $f⊣g$ to mean that $f$ is left-adjoint and $g$ is right-adjoint.

Properties

Adjunctions compose. Equivalence of categories preserve adjunctions. Equivalence of functors preserve adjunctions. The left adjunction of a morphism is unique up to equivalence. If $f\cong g$ and $f$ is an equivalence then $g$ is also an equivalence.