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Let $\mathcal{C}$ be a $2\!-$category and let $(F,G,\eta,\varepsilon)$, $(F',G',\eta',\varepsilon')$ be two paralell adjunctions $A-\!\!\!\rightharpoonup B$ in $\mathcal{C}$. Let $F\xrightarrow{\ \sigma\ }F'$, $G'\xrightarrow{\ \tau\ }G$ be $2\!-$morphisms. If $\mathcal{C}=\mathbf{Cat}$, so that we deal with ordinary adjoint functors, it is proven in Categories for the Working Mathematician, that any of the four equalities

$$\tau=(1_G\bullet\varepsilon')\circ(1_G\bullet\sigma\bullet 1_{G'})\circ(\eta\bullet1_{G'})\,,$$

$$\sigma=(\varepsilon\bullet 1_{F'})\circ(1_F\bullet\sigma\bullet 1_{F'})\circ(1_{F}\bullet\eta')\,,$$

$$\varepsilon\circ(1_F\bullet\tau)=\varepsilon'\circ(\sigma\bullet 1_{G'})\,,$$

$$(1_G\bullet\sigma)\circ\eta=(\tau\bullet 1_{F'})\circ\eta'\,,$$

are equivalent.

Does this result hold in the more general setting of arbitrary $2\!-$categories and adjunctions? If so, how can I prove it?

Unfortunately, Mac Lane uses the definition of adjunctions via isomorphisms of $\operatorname{Hom}$-functors, so that the proof does not work in this more general context.

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Use the 2-categorical Yoneda embedding to reduce to the case of $\mathfrak{Cat}$. –  Zhen Lin Jul 9 at 18:16

1 Answer 1

Actually, the definition of adjunctions via natural isomorphisms of Hom-functors works in any $2$-category. Specifically, $f\leftrightarrows g$ are adjoint with unit $1\stackrel\eta\Rightarrow fg$ and counit $gf\stackrel\epsilon\Rightarrow 1$ if and only if we have a family of isomorphisms $[fX,Y]\cong[X,gY]$ that is natural in the "$E$-objects" $E\stackrel X\rightarrow C$ of $C$ and $E\stackrel Y\rightarrow D$ of $D$, for every object $E$ of the $2$-category.

One may wonder what kind of Functors $[fX,Y]$ and $[X,gY]$ are, and also between what Categories are they Functors, but one should instead consider them to be just "syntactic sugar" to sweeten one's manipulations of Kan lifts and Kan extensions.

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