I'm having trouble with getting started with this problem in Category Theory in Context, so any help would be appreciated.
Problem:
So the set up is we have isomorphism natural in both $c$ and $d$ such that $D(Uc, d) \cong C(c, Ld)$ and $C(Rd, c) \cong D(d, Uc)$ for all $c \in C$ and $d \in D$.
We want to show there is isomorphism that is natural such that $C(LUc, c') \cong C(c, RUc')$.
I'm having a hard time getting started with the proof, so if anyone could help me get started, I would appreciate that very much. Thanks!
Alright, I was just being stupid here.
What we have instead is
$C(Ld,c) \cong D(d, UC)$ and $D(Uc, d) \cong C(c, Rd)$.
Then note $C(LUc, c') \cong D(Uc, Uc')$ naturally, and $D(Uc, Uc') \cong C(c, RUc')$ naturally.