A local approach to adjunctions in a 2-category

Question

I have trouble understanding the following pragraph in Stephen Lack's A 2-Categories Companion. You can find it near the bottom of page 8, here is a link the article.

The local approach to adjunctions also works well here, provided that one uses generalized objects rather than objects. For any 1-cells $a:X\to A$ and $b:X\to B$, there is a bijection between 2-cells $fa\to b$ and 2-cells $a\to ub$. One now has naturality with respect to both 1-cells $x:Y\to X$, and 2-cells $a\to a'$ or $b\to b'$. This local-global correspondence can be proved more or less as in the usual case, or it can be deduced from the usual case using a suitable version of the Yoneda lemma. In fact the global-to-local part...

I am interested in the local-to-global part. What do they mean when they say that $f \dashv u$ can be deduced with help of a suitable version of the Yoneda lemma? What is the version of the Yoneda lemma I have to use?

Answer 1

Lexponent has given a proof that spells out the details, but perhaps it would also be useful to give a higher level perspective of what's going on.

The claim Lack is essentially making is this: given 1-cells $f : A \to B$ and $g : B \to A$ in a 2-category $\mathcal K$, the following are equivalent.

1. $f \dashv g$ in $\mathcal K$.
2. For all objects $X \in \mathcal K$, we have $\mathcal K[X, f] \dashv \mathcal K[X, g]$ in $\mathbf{Cat}$ 2-natural in $X$.

If you spell out (3), you recover Lack's description of the "local approach" to adjunctions in a 2-category: for each object $X$ and 1-cells $a : X \to A$ and $b : X \to B$, there is a bijection between 2-cells $f a \Rightarrow b$ and 2-cells $a \Rightarrow g b$. 2-naturality comes for free, since the bijections are induced by whiskering with the unit, which arises from taking $X = A$, $a = 1_A$, $b = f$; and the counit, which arises from taking $X = B$, $a = g$, $b = 1_B$.

The key observation in establishing the equivalence of these two definitions is that (2) is an adjunction in the 2-presheaf category $[\mathcal K^\circ, \mathbf{Cat}]$. The equivalence between then follows from the 2-Yoneda lemma.

(1) ⇒ (2). The 2-Yoneda embedding $\mathcal K[X, {-}]$ of $X$ is a 2-functor, and 2-functors preserve adjunctions.

(2) ⇒ (1). The 2-Yoneda lemma implies that the 2-Yoneda embedding $\mathcal K[X, {-}]$ of $X$ is locally fully faithful, and hence reflects adjunctions. Conceptually, this is because adjunctions $f \dashv g$ are defined in terms of 2-cells (the unit and the counit) and equations between them, which are respected by locally fully faithfulness (just as fully faithful functors preserve all diagrammatic definitions).

Answer 2

Let me provide a step-by-step explanation of how to deduce the adjunction $f \dashv u$ using the 2-categorical Yoneda lemma.

1. Start with the given adjunction $f \dashv u$ in the 2-category $\mathscr{K}$.

2. Consider the representable 2-functor $\mathscr{K}(X, -) : \mathscr{K} \rightarrow \mathrm{Cat}$, where $X$ is any object in $\mathscr{K}$.

3. Apply the 2-categorical Yoneda lemma to the representable 2-functor $\mathscr{K}(X, -)$. This tells us that, for any 1-cell $a : X \rightarrow A$ and any 1-cell $b : X \rightarrow B$, there is a bijection between 2-cells $f a \rightarrow b$ and 2-cells $a \rightarrow u b$, natural in both $a$ and $b$.

4. Now, we want to deduce the adjunction $f \dashv u$ from this bijection. To do this, consider the 1-cells $a : X \rightarrow A$ and $b : X \rightarrow B$ as fixed.

5. By the naturality in both $a$ and $b$, we can find two natural transformations, $\alpha: \mathscr{K}(X, f a) \rightarrow \mathscr{K}(X, b)$ and $\beta: \mathscr{K}(X, a) \rightarrow \mathscr{K}(X, u b)$, such that the bijection between 2-cells $f a \rightarrow b$ and 2-cells $a \rightarrow u b$ corresponds to the bijection between components of $\alpha$ and $\beta$.

6. In other words, for any $x : Y \rightarrow X$, the component of $\alpha$ at $x$, denoted as $\alpha_x$, is a morphism from $f(ax) \rightarrow bx$ in $\mathscr{K}(Y, B)$, and the component of $\beta$ at $x$, denoted as $\beta_x$, is a morphism from $ax \rightarrow u(bx)$ in $\mathscr{K}(Y, A)$.

7. Since these natural transformations are given by the 2-categorical Yoneda lemma, we can conclude that $f \dashv u$ holds in the 2-category $\mathscr{K}$, as the bijection between 2-cells $f a \rightarrow b$ and 2-cells $a \rightarrow u b$ given by the Yoneda lemma is exactly what we need for a local description of the adjunction.

Hence, the 2-categorical Yoneda lemma provides a bijection between 2-cells $f a \rightarrow b$ and 2-cells $a \rightarrow u b$, which is natural in both $a$ and $b$. This bijection allows us to deduce the adjunction $f \dashv u$ by relating the components of the natural transformations given by the Yoneda lemma to the morphisms in the 2-category $\mathscr{K}$.