If is bijective show that is unique.
Suppose that and are inverses of . Let . We need to show that for all . So for all , thus is unique?
Let such that for all . To show that we need to show to things, (i) they have the same domain (ii) for all in domain. The first part is straightforward, it is given that have domain . Now let . Then since is bijective it means for some . Thus, . And thus these are the same functors.