# The Unapologetic Mathematician

## Mathematics for the interested outsider

When I started in on adjoint functors, I gave the definition in terms of a bijection of hom-sets. Then I showed that we can also specify it in terms of its unit and counit. Both approaches (and their relationship) generalize to the enriched setting.

Given a functor $F:\mathcal{C}\rightarrow\mathcal{D}$ and another $G:\mathcal{D}\rightarrow\mathcal{C}$, an adjunction is given by natural transformations $\eta:1_\mathcal{C}\rightarrow G\circ F$ and $\epsilon:F\circ G\rightarrow1_\mathcal{D}$. These transformations must satisfy the equations $(1_G\circ\epsilon)\cdot(\eta\circ1_G)=1_G$ and $(\epsilon\circ1_F)\cdot(1_F\circ\eta)=1_F$. By the weak Yoneda Lemma, this is equivalent to giving a $\mathcal{V}$-natural isomorphism $\phi_{C,D}:\hom_\mathcal{D}(F(C),D)\rightarrow\hom_\mathcal{C}(C,G(D))$.

Indeed, a $\mathcal{V}$-natural transformation in this direction must be of the form $\phi_{C,D}=\hom_\mathcal{C}(\eta_C,1_{G(D)})\circ G_{F(C),D}$, and one in the other direction must be of the form $\varphi_{C,D}=\hom_\mathcal{D}(1_{F(C)},\epsilon_D)\circ F_{C,G(D)}$. The equations $\phi\circ\varphi=1$ and $\varphi\circ\phi=1$ are equivalent, by the weak Yoneda Lemma, to the equations satisfied by the unit and counit of an adjunction.

The $2$-functor $\mathcal{V}\mathbf{-Cat}\rightarrow\mathbf{Cat}$ that sends an enriched category to its underlying ordinary category sends an enriched adjunction to an ordinary adjunction. The function underlying the $\mathcal{V}$-natural isomorphism $\phi$ is the bijection of this underlying adjunction.

As we saw before, a $\mathcal{V}$-functor $G$ has a left adjoint $F$ if and only if $\hom_\mathcal{D}(D,F(\underline{\hphantom{X}}))$ is representable for each $D\in\mathcal{D}$. Also, an enriched equivalence is an enriched adjunction whose unit and counit are both $\mathcal{V}$-natural isomorphisms. Just as for ordinary adjunctions, we have transformations between enriched adjunctions, a category of enriched adjunctions between two enriched categories, enriched adjunctions with parameters, and so on.