I'm working through the section on adjoint functors in Borceaux's Categorical Algebra. If $F: C \to D$ is a functor, and $d \in |D|$, he defines a reflection of $d$ across $F$ to be a pair $(L(d), \eta_d)$, where $L(d) \in |C|$ and $\eta_d: d \to FL(d)$ such that for any arrow $\phi: d \to F(c)$, there exists a unique $\phi^*:L(d) \to c$ such that $F(\phi^*)\circ \eta_d = \phi$. (In other words, $L(d), \eta_{d}$ is an initial object in the category $d \downarrow F$ of arrows under $d$.) The dual notion is a coreflection.
Given functors $F: C \to D$ and $G: D \to C$, he gives some equivalent characterizations for an adjunction $G \vdash F$. The following two involve the unit and counit:
- There exists a natural transformation $\eta: \text{Id}_D \to F \circ G$ such that for each $d \in |D|$, $G(d)$ is a reflection of $d$ along $F$.
- There exists a natural transformation $\epsilon: G \circ F \to \text{Id}_C$ such that for each $c \in |C|$, $F(c)$ is a coreflection of $c$ along $G$.
I'm looking for a direct proof of (1) $\implies$ (2). I'm 75% of the way there. So far I have: Let $\epsilon$ take $c \in |C|$ to ${\text{Id}_{F(c)}}^*: GF(c) \to c$ (with the asterisk notation as defined above). I've checked that it's natural using the reflection property in (1). Now I want to show that for any $c \in |C|$, we have that $F(c), \epsilon_c$ is a coreflection: i.e. given any $d \in |D|$ and $\phi:G(d) \to c$, there is a unique $\phi^\star: d \to F(c)$ such that ${\text{Id}_{F(c)}}^* \circ G(\phi^\star) = \phi$.
I'm stuck on this. I'd like to prove it as a gateway to proving all the usual adjunction stuff (i.e. I'd like to prove it on its own without taking for granted all the usual stuff about adjunctions).
Any tips?
Edit: I see that the dual question is asked and answered in this post. I will try to work it out from there.