# When two morphisms are dual to each other in a 2 category?

Here is one famouse approach of defining Adjoint functors:

We say $F: D\rightarrow C$ is left adjoint to $G:C \rightarrow D$ or equivalently $G$ is right adjoint to $F$ if $$C\left(FY,X\right) \cong D\left(Y,GX\right)$$ is natural in both X and Y.

If I am right, an adjunction is such a system of $\left(C,D,F,G,\mu,\theta\right)$ where $\mu$ and $\theta$ are natural transformations or 2-morphism in the category of categories. Now my question is how the more general notion of adjointion in any 2-category can be formulated. As IAs I have learned from the answer bellow, two morphisms in a 2 category form an adjunction if they are dual to each other. I will be very thankful if one could give a link to some relevant readings or spell out the defenition of adjunction in a 2-category.

Thank you.

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## 2 Answers

The correct definition of adjunction in a general (strict) 2-category is the algebraic one, which I repeat below:

Definition. Let $\mathfrak{K}$ be a (strict) 2-category. An adjunction in $\mathfrak{K}$ is a quadruple $(f, g, \eta, \epsilon)$ where $f : C \to D$ and $g : D \to C$ are 1-cells, $\eta : \textrm{id}_C \Rightarrow g f$ and $\epsilon : f g \Rightarrow \textrm{id}_D$ are 2-cells, these satisfying the triangle identities: \begin{align} \epsilon f \bullet f \eta & = \textrm{id}_f \\ g \epsilon \bullet \eta g & = \textrm{id}_g \end{align}

Here, $\bullet$ denotes the vertical composition of 2-cells, and I use the usual abbreviations for whiskering. If $\mathfrak{K}$ is only a bicategory then one has to insert coherence isomorphisms in the appropriate places.

The nice thing about the algebraic definition is that it is manifestly preserved by all 2-functors; in particular, for any object $T$ in $\mathfrak{K}$, the induced functors on hom-categories $f_* : \mathfrak{K}(T, C) \to \mathfrak{K}(T, D)$ and $g_* : \mathfrak{K}(T, D) \to \mathfrak{K}(T, C)$ will be adjoint in the usual sense, with unit and counit induced from $\eta$ and $\epsilon$ in the obvious way.

As for duality: this notion is really something from monoidal categories, not 2-categories. But it is certainly true that a dual pair of objects is like an adjoint pair of functors.

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The question is not clear. But I am pretty sure that the nlab article on adjunctions contains everything you want.

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Thank you. I will spend some time with that article. But, what is meaning of duality in adjoint – Hooman Mar 11 '13 at 14:57