I know the result should be true, but I saw this very interesting proof here:
The idea is to find a category (the twisted arrow category), having adjunctions to $\mathcal{C}$ and $\mathcal{C}^{\mathrm opp}$, then by some abstract homotopy theory we have the equivalence.
My question is, how could we find the adjunctions? (Namely, how to solve Exercise 1.5?)