Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair? In my mind $F\dashv G$ ...
(There are several dialects of 2-categorical language. Mine is the one where everything is weak by default. You might need to insert the word "weak" or "pseudo" (or even "strong", if your default is ...