Motivating the definition of adjoint equivalence

Question

Recently I have been trying to convince myself that the most natural definition for equivalence between categories is the notion of adjoint equivalence rather than simply equivalence. Of course, every equivalence of categories can be "upgraded" to an adjoint equivalence, but conceptually I feel that in many ways the notion of adjoint equivalence is more natural.

Adjoint equivalences have the following advantages over (non-adjoint) equivalences as a notion of equivalence:

1. For a functor $F: \mathcal{C} \to \mathcal{D}$, any functor $G:\mathcal{D} \to\mathcal{D}$ which is forms an adjoint equivalence with $F$ is unique up to unique isomorphism (in fact, this holds for any pair of adjoint functors). On the other hand, if we require that $F,G$ is just an equivalence, then the isomorphism need not be unique.
2. If $F: \mathcal{C} \to \mathcal{D}$ is fully faithful and essentially surjective then the only sensible way to construct $G: \mathcal{D}\to \mathcal{C}$ such that $F,G$ is an equivalence actually yields an adjoint equivalence.

I have a feeling that the 2-category structure on the category of ((locally) small) categories $\mathrm{Cat}$ is behind this phenomenon: non-adjoint equivalences are equivalences from the point of view of vertical composition, and the triangle identities for an adjunction ensure that $F,G$ are an equivalence from the point of view of horizontal composition.

However, this isn't quite satisfactory. If I wanted to naively write down what $F:\mathcal{C} \to \mathcal{D}:G$ being horizontal-inverses to one another means, I would write down the following: There exists natural transformations $$\eta: id_{\mathcal{C}}\Rightarrow GF ,\overline{\eta}:GF \Rightarrow id_{\mathcal{C}}$$ $$\varepsilon: FG \Rightarrow id_{\mathcal{D}} ,\overline{\varepsilon}: id_{\mathcal{D}}\Rightarrow FG$$such that $$\overline{\eta} \ast\eta =\eta \ast\overline{\eta} = id_{\mathcal{C}}$$ $$\overline{\varepsilon} \ast\varepsilon =\varepsilon \ast\overline{\varepsilon} = id_{\mathcal{D}}$$ However, unpacking these definitions just gives $$\eta_x \circ \overline{\eta}_x=id_x, \varepsilon_x \circ \overline{\varepsilon}_x=id_x $$ so this simply boils down to equivalence.

My questions are the following:

1. Am I right in thinking that adjoint equivalence is the "right" notion of equivalence, rather than non-adjoint equivalence?
2. If so, is this ultimately due to the 2-category structure on $\mathrm{Cat}$?
3. In this context, how can one motivate the triangle identities? Perhaps there are many laws we want $F,G$ to satisfy and it turns out that $\eta,\varepsilon$ being isomorphisms and satisfying the triangle identities is sufficient for all of these laws holding (in much the same way that the pentagon identity in a bicategory is a sufficient condition for all of the infinitely-many association laws which one might desire)

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Here I use $\circ$ to refer to vertical composition and $\ast$ to refer to horizontal composition

Answer 1

I think I have an answer to this now. The "right" notion of equivalence of categories (in my opinion) consists of natural transformations $$\eta: id_{\mathcal{C}} \Rightarrow GF,\overline{\eta}: GF \Rightarrow id_{\mathcal{C}} $$ $$\varepsilon: id_{\mathcal{D}} \Rightarrow FG,\overline{\varepsilon}: FG \Rightarrow id_{\mathcal{D}}$$ subject to the following relations: Consider the functors $id_{\mathcal{C}},id_{\mathcal{D}}$ along with all the iterated composites of $F,G$ (i.e. $F,G,FG,GF,FGF,GFG$, etc). Let $F_i=FG...F$ (or $FG...G$) of length $i$ with $F_0=id_{\mathcal{C}}$ and $G_i=GF...G$ (or $GF...F$ etc) of length $i$ with $G_0=id_{\mathcal{D}}$. Let $$\eta_{i,j}=id_F \ast \ldots \ast\eta \ast\ldots \ast id_F: F_i\Rightarrow F_{i+2}$$ have $\eta$ in the $j$-th position (where $j$ is even) and define $$\overline{\eta}_{i,j}:F_i\Rightarrow F_{i-2} \text{ (for $j$ even)}$$ $$\eta_{i,j}':G_i\Rightarrow G_{i+2} \text{ (for $j$ odd)}$$ $$\overline{\eta}_{i,j}':G_i\Rightarrow G_{i-2} \text{ (for $j$ odd)}$$ $$\varepsilon_{i,j}:F_i\Rightarrow F_{i+2} \text{ (for $j$ odd)}$$ $$\overline{\varepsilon}_{i,j}:F_i\Rightarrow F_{i-2} \text{ (for $j$ odd)}$$ $$\varepsilon_{i,j}':G_i\Rightarrow G_{i+2} \text{ (for $j$ even)}$$ $$\overline{\varepsilon}_{i,j}':G_i\Rightarrow G_{i+2} \text{ (for $j$ even)}$$ similarly. We require that any two sequences of the above natural transformations from $F_a,F_b$ have to be equal (after vertical composition) and the same for $G_a,G_b$.

Claim:

The data $(F,G,\eta,\varepsilon,\overline{\eta},\overline{\varepsilon})$ forms an equivalence in the above sense if and only if $\eta,\varepsilon$ are natural isomorphisms with $\eta^{-1}=\overline{\eta},\varepsilon^{-1}=\overline{\varepsilon}$ and $(F,G,\eta,\varepsilon)$ is an adjoint equivalence.

Sketch proof: ($\Rightarrow$): This direction is easy to see - the relations $\eta\circ\overline{\eta}=\overline{\eta}\circ\eta=id_{\mathcal{C}},\varepsilon\circ\overline{\varepsilon}=\overline{\varepsilon}\circ\varepsilon=id_{\mathcal{D}}$ and the triangle identities for adjunctions are both instances of the laws given above.

($\Leftarrow$): It is not hard to convince oneself that for any sequence of the given natural transformations from $F_a$ to $F_b$ (or $G_a$ to $G_b$), the corresponding commutative diagram can be "filled in" by triangles, and hence that we need only check relations of a triangular shape. Now consider how the insertion and deletion of $GF$ or $FG$ by the natural transformations overlap. The only nontrivial cases are when $GF$ or $FG$ have been inserted/deleted in exactly the same position, or when the $F$ in an inserted $GF$ overlaps with the $F$ in a deleted $FG$ (or some permutation of $F$, $G$, "inserted", "deleted" etc.) But these are exactly the laws for equivalence/adjointness respectively! $\square$