In my category theory course, we've defined an equivalence of categories to be a pair of adjoint functors $(L, R)$, i.e. $\text{Hom}_{\mathcal D}(LC, D) \cong \text{Hom}_{\mathcal C}(C, RD)$ in a natural way, such that $L$ and $R$ are fully faithfull. An equivalent definition is that $(L, R)$ is an adjunction and the unit and the counit $\eta, \varepsilon$ are natural isomorphisms.
It follows from the second definition, by reversing $\eta$ and $\varepsilon$, that $(R, L)$ is still an adjunction therefore $R, L$ are both left and right adjoint. Indeed
$$\text{Hom}_{\mathcal D}(D, LC) \overset{R}{\longrightarrow} \text{Hom}_{\mathcal C}(RD, RLC)\overset{\eta^{-1}_C \circ -}{\longrightarrow}\text{Hom}_{\mathcal C}(RD, C)$$
and we can go the other way around by using $L$ and $\varepsilon_D^{-1}$. Now I have some troubles to show the fact that $(R, L)$ is still an adjunction but only by using the first definition of equivalence of category, i.e. the fully faithfullness of $R, L$. Does anyone know how to do that ?
Best Answer
Here's a proof:
Observe that $L$ and $R$ being full and faithful is equivalent to there being natural bijections: $${\rm Hom}_{\mathcal{C}}(C,C') \cong {\rm Hom}_{\mathcal{D}}(LC,LC') \ \ (1), \ \ \ \ {\rm Hom}_{\mathcal{D}}(D,D') \cong {\rm Hom}_{\mathcal{C}}(RD,RD') \ \ (2).$$ Then we have the following natural bijection: \begin{equation} \begin{split} {\rm Hom}_{\mathcal{D}}(D,LC) & \cong {\rm Hom}_{\mathcal{C}}(RD,RLC) \ \text{ by (1),}\\ &\cong {\rm Hom}_\mathcal{D}(LRD,LC) \ \text{ since $L \dashv R$,}\\ &\cong {\rm Hom}_\mathcal{C}(RD,C) \ \text{ by (2).} \end{split} \end{equation} Thus $R \dashv L$.