The Unit and Counit of an Adjunction

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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