Note: for a category C, we write [X, Y]_C for the hom-class of maps from X to Y.

An adjunction is a pair of functors F : C \rightarrow D and G : D \rightarrow C with a certain relationship. There are many texts expositing adjunctions, but I have not seen one covering what I think of as their fundamental theorem. This is regrettable, since I think many people would appreciate the simplifying effect this result has on the theory of adjunctions in \text{Cat}. My goal today is to cover this theorem using coend calculus.

Adjunctions in a nutshell. First, an example. Let \text{Set} be the category of sets and let R \text{-mod} be the category of R-modules for a ring R. There are functors F : \text{Set} \rightarrow R \text{-mod} sending a set X to \oplus_{x \in X} R and a functor G : R \text{-mod} \rightarrow \text{Set} sending an R-module M to its underlying set. There is a natural way of going back and forth between maps of R-modules F(X) \rightarrow M and maps of sets X \rightarrow M. For a map of R-modules, f: F(X) \rightarrow M, we get a map of sets by first applying G (this really does nothing) and then precomposing with a map \eta_X of sets X \rightarrow G(F(X)) sending x to the element (a_y)_{y \in X} in the underlying set of F(X) where a_x = 1 and a_y =0 for y \neq x. The resulting map is X \stackrel{\eta_X}{\rightarrow} G(F(X)) \stackrel{G(f)}{\rightarrow} G(M). For a map of sets f: X \rightarrow G(M), we get a map F(X) \stackrel{f}{\rightarrow} F(G(M)) \stackrel{\epsilon_M}{\rightarrow} M where \epsilon_M: F(G(M)) \rightarrow M sends (a_x)_{x \in X} to \sum_{x \in X} a_X, where the sum is taken in M. What we have here is a natural correspondence between hom sets [F(X), M]_{R \text{-mod}} and [X, G(M)]_{\text{Set}}. To go one way we apply G and then precompose with \eta_X (kind of like correcting for an error, since F and G are not inverses), and to go the other way we apply F and then postcompose with \epsilon_M. This hints at the definition of an adjoint:

Definition: Let F : C \rightarrow D and G : D \rightarrow C be functors. We say that F is adjoint to G, or that F and G are adjoint, if there is a natural isomorphism \Phi : [ F-, -]_D \rightarrow [-, G-]_C between the functors [ F-, -]_D, [-, G-]_C : C^{op} \times D \rightarrow \text{Set}.

The example also hints at what I call the fundamental theorem of adjunctions in the 2-category \text{Cat}:

Theorem: Let C and D be categories and F and G be functors. Then
(i) [[F-, -], [-, G-]]_{[C^{op} \times D, \text{Set}]} and [1_C, GF]_{[C^{op}, C^{op}]} are naturally isomorphic.
(ii) [[-, G-], [F-, -]]_{[C^{op} \times D, \text{Set}]} and [FG, 1_D]_{[D, D]} are naturally isomorphic.

Proof: These are dual; we prove the first. For the isomorphism itself, we have a series of natural isomorphisms:

Screen Shot 2019-08-20 at 9.47.23 PM.jpg

This finishes the proof.

Using coend calculus like that is nice, but we are also interested in an explicit description of the isomorphisms \alpha : \text{Nat}_{[C^{op} \times D, \text{Set}]} ([F-, -], [-, G-]) \rightarrow \text{Nat}_{[C^{op}, C^{op}]}(1_C, GF) and \beta : \text{Nat}_{[C^{op}, C^{op}]}(1_C, GF) \rightarrow  \text{Nat}_{[C^{op} \times D, \text{Set}]} ([F-, -], [-, G-])! To achieve this description, notice that the following diagram commutes for each X \in \text{Obj}(C) and each Y \in \text{Obj}(D):

Screen Shot 2019-08-20 at 9.48.31 PM.jpg

Take a natural transformation \eta : 1 \rightarrow GF. For each X, define a natural transformation H_{X} : [FX, -] \rightarrow [X, G(-)] such that (H_{X})_Y : [FX, Y] \rightarrow [X, GY] sends f to G(f) \circ \eta_X, to (H_{X})_Y. Define a natural transformation H : [[F(-), -]_D, [-, G(-)]_{C}]_{[C^{op} \times D, \text{Set}]} where H_{X, Y} = (H_X)_Y. The composition

Screen Shot 2019-08-20 at 9.27.37 PM

gives the following diagram of mappings:

Screen Shot 2019-08-20 at 10.03.48 PM.jpg

As this holds for each X \in \text{Obj}(C) and each Y \in \text{Obj}(D), we have \alpha(\eta) = H.

Next take a natural transformation H : [[F(-), -]_D, [-, G(-)]_{C}]_{[C^{op} \times D, \text{Set}]}. For each X, define a natural transformation H_{X} : [FX, -] \rightarrow [X, G(-)] such that (H_{X})_Y = H_{X, Y}. Define a natural transformation \eta : 1_C \rightarrow GF such that \eta_X = (H_X)_{FX}(1_{FX}). The composition

Screen Shot 2019-08-20 at 9.27.37 PM

gives the following diagram of mappings:

Screen Shot 2019-08-20 at 10.04.08 PM.jpg

As this holds for each X \in \text{Obj}(C) and each Y \in \text{Obj}(D), \beta(H) = \eta.

Hence \alpha sends a natural transformation \eta : 1_C \rightarrow GF to the natural transformation H : [F-, -] \rightarrow [-, G-] where H_{X, Y}(f)= Gf \circ \eta_X for X \in \text{Obj}(C), Y \in \text{Obj}(D), and f : FX \rightarrow Y, and \beta sends a natural transformation H : [F(-), -]_D \rightarrow [-, G(-)]_{C} to the natural transformation \eta : 1_C \rightarrow GF such that \eta_X = H_{X, FX} (1_{FX}).

This theorem puts us in a place where we can prove other results about adjoints much more easily:

Theorem: The following are equivalent:

(i) The functors [F-, -]_D : C^{op} \times D \rightarrow \text{Set} and [-, G-]_C : C^{op} \times D \rightarrow \text{Set} are naturally isomorphic.
(ii)(a) (Universal Morphisms Definition) There is a natural transformation \eta : 1_{C} \rightarrow G F such that, for each X \in \text{Obj}(C), each Y \in \text{Obj}(D), and each morphism f : X \rightarrow GY in C, there is a unique morphism g : FX \rightarrow Y in \text{Mor}(D) such that Gg \circ \eta_X =f.
(ii)(b) (Universal Morphisms Definition) There is a natural transformation \epsilon : FG \rightarrow 1_{D} such that, for each Y \in \text{Obj}(D), each X \in \text{Obj}(C), and each g : FX \rightarrow Y in D, there is a unique morphism f : X \rightarrow GY in C such that \epsilon_Y \circ Ff = g.
(iii) (The Unit Counit Definition) There are natural transformations \epsilon : FG \rightarrow 1_{D} and \eta : 1_{C} \rightarrow G F such that the following compositions are the identity morphism, for each X \in \text{Obj}(C) and Y \in \text{Obj}(D).

Screen Shot 2019-08-20 at 9.52.01 PM.jpg

Proof: (i) \implies (ii)(a). Suppose that the functors [F-, -]_D : C^{op} \times D \rightarrow \text{Set} and [-, G-]_C : C^{op} \times D \rightarrow \text{Set} are naturally isomorphic, and take a natural isomorphism H : [F-, -]_D \rightarrow [-, G-]_C. Let \eta : 1_C \rightarrow GF be the corresponding natural transformation. Take X \in \text{Obj}(C), Y \in \text{Obj}(D), and f : X \rightarrow GY. There is a unique g : FX \rightarrow Y such that Hg = f, since H is a natural isomorphism. But Hg = Gg \circ \eta_X by the fundamental theorem. This shows (ii)(a).

(ii)(a) \implies (i). Suppose that there is a natural transformation \eta : 1_{C} \rightarrow G F such that, for each X \in \text{Obj}(C), each Y \in \text{Obj}(D), and each morphism f : X \rightarrow GY in C, there is a unique morphism g : FX \rightarrow Y in \text{Mor}(D) such that Gg \circ \eta_X =f. Let H : [F-, -]_D \rightarrow [-, G-]_C be the natural transformation corresponding to \eta. Take X \in \text{Obj}(C) and Y \in \text{Obj}(D). H_{X, Y}g = Gg \circ \eta_X for each g : X \rightarrow GY, so that, for each f \in [X, GY]_C, there is a unique g \in [FX, Y]_D such that H_{X, Y}g = Gg \circ \eta_X = f. This shows (i).

(i) \implies (ii)(b). Suppose that the functors [F-, -]_D : C^{op} \times D \rightarrow \text{Set} and [-, G-]_C : C^{op} \times D \rightarrow \text{Set} are naturally isomorphic, and take a natural isomorphism H : [-, G-]_C \rightarrow [F-, -]_D. Let \epsilon : FG \rightarrow 1_D be the corresponding natural transformation. Take X \in \text{Obj}(C), Y \in \text{Obj}(D), and g : FX \rightarrow Y. There is a unique f : X \rightarrow GY such that Hf = g, since H is a natural isomorphism. But Hf = \epsilon_Y \circ Ff by the fundamental theorem. This shows (ii)(b).

(ii)(b) \implies (i). Suppose that there is a natural transformation \epsilon : FG \rightarrow 1_{D} such that, for each X \in \text{Obj}(C), each Y \in \text{Obj}(D), and each morphism g : FX \rightarrow Y in D, there is a unique morphism f : X \rightarrow GY in \text{Mor}(D) such that \epsilon_Y \circ Ff \circ = g. Let H : [-, G-]_C \rightarrow [F-, -]_D be the natural transformation corresponding to \eta. Take X \in \text{Obj}(C) and Y \in \text{Obj}(D). H_{X, Y}f = \epsilon_Y \circ Ff for each f : X \rightarrow GY, so that, for each g \in [FX, Y]_C, there is a unique f \in [X, GY]_D such that H_{X, Y}f = \epsilon_Y \circ Ff = g. This shows (i).

(i) \implies (iii). Suppose that the functors [F-, -]_D : C^{op} \times D \rightarrow \text{Set} and [-, G-]_C : C^{op} \times D \rightarrow \text{Set} are naturally isomorphic, and take a natural isomorphism H : [F-, -]_D \rightarrow [-, G-]_C. Let \eta : 1_C \rightarrow GF be the natural transformation corresponding to H and let \epsilon : GF \rightarrow 1_D be the natural transformation corresponding to H^{-1}. For X \in \text{Obj}(C), we have
Screen Shot 2019-08-20 at 9.58.44 PM.jpg
and
Screen Shot 2019-08-20 at 9.58.58 PM.jpg
(iii) \implies (i). Suppose (iii) holds. Let E : [F-, -] \rightarrow [-, G-] be the natural transformation induced by \epsilon : FG \rightarrow 1_D, and let H : [-, G-] \rightarrow [F-, -] be the natural transformation induced by \eta : 1_C \rightarrow GF.

Take Y \in \text{Obj}(D). Write (EH)_{-, Y} : [-, GY] \rightarrow [-, GY] for the induced natural transformation. The following diagram commutes by the Yoneda lemma:
Screen Shot 2019-08-20 at 9.59.32 PM.jpg
So (EH)_{X, Y} (f) = [f, GY] \circ (EH)_{GY, Y} (1_{GY})=  (EH)_{GY, Y}(1_{GY}) \circ f = f. So (EH)_{-, Y} is the identity natural transformation. Hence $EH$ is the identity natural transformation.

Take Y \in \text{Obj}(D). Write (HE)_{X, -} : [FX, -] \rightarrow [FX, -] for the induced natural transformation. The following diagram commutes by the Yoneda lemma:
Screen Shot 2019-08-20 at 10.00.31 PM.jpg
So (HE)_{X, Y} (f) = [FX, f] \circ (HE)_{X, FX} (1_{FX}) = f \circ (HE)_{X, FX}(1_{FX}) = f. So (HE)_{X,-} is the identity natural tarnsformation. Hence HE is the identity natural transformation.

It follows that H and E are inverse natural transformations, and therefore natural isomorphisms. This shows (i).

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