A trace-monoidal category is equipped with a trace:
\[ Tr \colon C(a \otimes x, b \otimes x) \to C(a, b) \]
You might be tempted to define a cotrace as:
\[ C(a, b) \to C(a \otimes x, b \otimes x) \]
but it's trivial (functoriality of the tensor product).

Except when you generalize hom-sets to profunctors. A profunctor equipped with a (cotrace?) natural transformation:
\[ P(a, b) \to P(a \otimes x, b \otimes x) \]
is called a Tambara module.

Conversely, the trace generalizes to co-Tambara modules:
\[ P(a \otimes x, b \otimes x) \to P(a, b)\]