symmetric monoidal traces

[Michael Shulman <shulman@math.uchicago.edu>]

Hi all,

Can someone please point me to whatever categorical references exist
regarding the canonical trace for dualizable objects in a symmetric
monoidal category?  I am particularly interested in (1) elementary
expositions accessible to non-category-theorists and (2) any proofs of
its uniqueness subject to various conditions.

I know there are many references on traced monoidal categories,
particularly with applications to computer science, but right now I am
only interested in the symmetric monoidal trace.  I also know that
there are various reinventions/expositions of the notion in, for
example, the topological literature (e.g. Dold-Puppe), but I would
like an exposition not tied to any particular application.  Finally, I
know that the Joyal-Street-Verity paper "Traced Monoidal Categories"
proves that the canonical symmetric (or, more precisely, balanced)
monoidal trace is "universal" in that any traced monoidal category can
be embedded in one equipped with the canonical trace, but as far as I
can tell this need not determine the canonical trace uniquely.

Thanks!!
Mike