A computable structure A is computably categorical if for all computable copies B, there exists a computable isomorphism between A and B. We can relativize this notion to a specific Turing degree d in the following way: a computable structure A is computably categorical relative to d if for all d-computable copies B, there exists a d-computable isomorphism between A and B. In this talk, we will discuss notable behaviors of this notion in the Turing degrees, such as its nonmonotonicity below 0'. We will also discuss which classes of structures can have a computable witness to these behaviors. This talk is based on work from https://arxiv.org/abs/2401.06641 and https://arxiv.org/abs/2505.15706.