I'm working on Exercise 1.2.25 from https://arxiv.org/pdf/1612.09375.pdf.
Parts (a) and (b) are straightforward as far as I understand, but I'm not sure how to do (c). I can define the functor $F$ on objects in at least two ways: $F((A,B))=F^A(B)$ or $F((A,B))=F_B(A)$. According to (b), these two agree, so we can choose either definition. But I don't know how to define $F$ on morphisms and how to use the second condition in (b). Further, jumping to the question of uniqueness, is there a standard technique to prove it?