# [Reference Request] The Definition of Adjoint Functors between dg-categories

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 $F\dashv G$ requires that there is a quasi-isomorphism between the cochain complext $\text{Hom}_B(Fx, y)$ and $\text{Hom}_A(x, Gy)$ for any $x\in \text{Obj}(A)$ and $y\in \text{Obj}(B)$. Can we make it more precise?

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It is not clear to me that there is such a thing as "the" definition. Probably different definitions are suitable for different purposes. For any monoidal category $V$ there is a $2$-category of $V$-enriched categories, $V$-enriched functors, and $V$-enriched natural transformations, and the unit-counit definition of an adjunction naturally generalizes to any $2$-category including this one. But the resulting notion of adjunction may be too restrictive...? –  Qiaochu Yuan Sep 23 '12 at 8:14
@Zhaoting Your second paragraph looks precise enough to me. –  Fernando Muro Sep 23 '12 at 9:34
I'll leave this as a comment, since I don't have an answer. Certainly "there is" means that you have chosen one naturally in $x$ and $y$. The subtle question is to understand how "natural" the choice must be. One should not expect it to be strictly functorial in $x$ and $y$, but rather functorial up to higher homotopies. –  Theo Johnson-Freyd Sep 23 '12 at 17:16
@Theo: Yes that's what I am worried about. –  Zhaoting Wei Sep 23 '12 at 19:16
You might find this n-lab page on adjoint functors in (oo,1)-categories helpful ncatlab.org/nlab/show/adjoint+(infinity,1)-functor. –  Sam Gunningham Sep 25 '12 at 5:05
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