Counit-unit adjunction

Collection

context	$F$ in ${\bf D}\longrightarrow{\bf C}$
context	$G$ in ${\bf C}\longrightarrow{\bf D}$
definiendum	$\langle\varepsilon,\eta\rangle$ in $F\dashv G$
inclusion	$\varepsilon, \eta$ … my nice nats $\left(F,G\right)$
for all	$X\in{\bf C}, Y\in{\bf D}$
postulate	$\varepsilon_{FY}\circ F(\eta_Y)=1_{FY}$
postulate	$G(\varepsilon_X)\circ \eta_{GX}=1_{GX}$

Elaboration

The pair $\langle\varepsilon,\eta\rangle$ being nice nats for $F$ and $G$ means

$\varepsilon:FG\xrightarrow{\bullet}1_{\bf C}$

$\eta:1_{\bf D}\xrightarrow{\bullet}GF$

Idea

Counit-unit adjunctions should be contrasted with my equivalence of categories, which is another special case of nice nats. In the case of equivalences, $\langle\varepsilon,\eta\rangle$ are isomorphisms

$\alpha$ in $FG\cong 1_{\bf C}$

$\beta$ in $1_{\bf D}\cong GF$

In the case of equivalence, we can go from a category ${\bf D}$ along $F$ (to the image of ${\bf D}$ in ${\bf C}$, call that “image 1”) and then back along $G$ (the image of “image 1” in ${\bf D}$, call it “image 2”) and find the same (${\bf D}$ and “image 2” are actually isomorphic). This possibility for invertibility means nothing was lost when passing from ${\bf D}$ to “image 1”.

In the case of an adjunction, not both nats are invertible. However, we need not go two times along a functor to invert! We already know about an left-invertibility relation of $\eta$ (either in the form $F(\eta_Y)$ or $\eta_{GX}$) once we go to the first image.

$\varepsilon_{FY}\circ F(\eta_Y)=1_{FY}$

$G(\varepsilon_X)\circ \eta_{GX}=1_{GX}$

There is also the combined case where you have an equivalence where the natural transformations are related in the sense of above - this is called an adjoint equivalence.

Inducing hom-set adjunctions

Say you're given an arrow $f$ from or to the images of one of the functors (in either ${\mathrm{Hom}}(FX,Y)$ or ${\mathrm{Hom}}(X,GY)$). We can now pre- or post-compose with arrows formed from $\eta$ and $\epsilon$, use the functors on arrows and thus algebraically find an image of $f$ in the other category.

Of course, each identity morphisms $1_{FX}:{\mathrm{Hom}}(FX,FX)$ in ${\bf C}$ corresponds to a component $\eta_X:{\mathrm{Hom}}(X,GFX)$ of $\eta:1_{\bf D}\xrightarrow{\bullet}GF$. And the claim here is that not only

$1_{FX}\leftrightarrow \eta_X$

or even

${\mathrm{Hom}}(FX,FX)\cong{\mathrm{Hom}}(X,GFX)$,

but in fact

${\mathrm{Hom}}(FX,Y)\cong{\mathrm{Hom}}(X,GY)$

It's not that hard do the construction in both directions, after you've written down the types of $\eta,\epsilon, F, G$ before you.

As universals

For another perspective relating to universal morphisms, see On universal morphisms (31.10.2014).

To monads

Having an adjoint functor pair really means you also got a nice pair of natural transofmrations (for which functors are only a conditions). Given any functor $G$ (w.l.o.g, say you're in ${\bf D}$ and the functor out of it is $G$), then if there is an $F$ so that $F\dashv G$, you got yourself a monad.

Theorems

It's important to note that as soon as (the fmap of one of) the adjoint functors are full and faithful, the adjunction provides and equivalence of categories.

Terminology/Notation

To remember the symbol of the counit and unit, maybe it helps to point out that $\varepsilon$ kinda looks like a $c$ and $\eta$ kinda looks like a turned around $u$.

The arrow $\eta:1_{\bf D}\xrightarrow{\bullet}GF$ called the “unit” (or “return”, in the programming world). Here a mnemonic I cam up with:

tfw oneitis returns, becomes your GF and wants the D

The functor $F$ in $F\dashv G$ is the left adjoint. Analogously, $G$ is the right adjoint functor.

Examples

power set/list monad, also list-monad ⇔ set to free monoid