The Unit and Counit of an Adjunction

The Unapologetic Mathematician

Let’s say we have an adjunction $latex Fdashv G:mathcal{C}rightarrowmathcal{D}$. That is, functors $latex F:mathcal{C}rightarrowmathcal{D}$ and $latex G:mathcal{D}rightarrowmathcal{C}$ and a natural isomorphism $latex Phi_{C,D}:hom_mathcal{D}(F(C),D)rightarrowhom_mathcal{C}(C,G(D))$.

Last time I drew an analogy between equivalences and adjunctions. In the case of an equivalence, we have natural isomorphisms $latex eta:1_mathcal{C}rightarrow Gcirc F$ and $latex epsilon:Fcirc Grightarrow1_mathcal{D}$. This presentation seems oddly asymmetric, and now we’ll see why by moving these structures to the case of an adjunction.

So let’s set $latex D=F(C’)$ like we did to show that an equivalence is an adjunction. The natural isomorphism is now $latex Phi_{C,F(C’)}:hom_mathcal{D}(F(C),F(C’))rightarrowhom_mathcal{C}(C,G(F(C’))$. Now usually this doesn’t give us much, but there’s one of these hom-sets that we know has a morphism in it: if $latex C’=C$ then $latex 1_{F(C)}inhom_mathcal{D}(F(C),F(C))$. Then $latex Phi_{C,F(C)}(1_{F(C)})$ is an arrow in $latex mathcal{C}$ from $latex C$ to $latex left[Gcirc Fright](C)$.

We’ll call this arrow $latex eta_C$. Doing this for every object $latex Cinmathcal{C}$…

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