I've seen some definitions of "right partial trace" and "left partial trace" in http://arxiv.org/abs/1103.1660, but these don't seem canonical in any way.

The motivation for this questions is that I'm thinking about topological quantum computing / modular tensor categories. It seems to me (from my limited physical understanding) like there should be a way to "discard" an object X, and go from a morphism $X \otimes A \to X \otimes A$ to one from $A \to A$. I can't really see what a canonical way of doing so would be, though.

First time posting a question here -- I hope this makes some sense!