This MO question asks for a geometric interpretation of the trace of a linear transformation. I'm wondering about a geometric interpretation of partial trace.

Given a linear transformation $f: X\otimes U\to Y\otimes U,$ the partial trace is a linear transformaton $\text{Tr}^U_{X,Y}\;\;(f):X\to Y$ satisfying certain properties. Basically, if you think of $f$ as a matrix with $|X|\times|Y|$-many $|U|\times|U|$ blocks, then $\text{Tr}(f)$ is given by taking the trace of each block.

So how can I visualize this operation? How can I tell a story about it, especially without resorting to a choice of bases?