  The goal of this paper is to construct trace maps for the six functor
formalism of motivic cohomology after Voevodsky, Ayoub, and
Cisinski-D\'{e}glise. We also construct an $\infty$-enhancement of such a trace
formalism. In the course of the $\infty$-enhancement, we need to reinterpret
the trace formalism in a more functorial manner. This is done by using
Suslin-Voevodsky's relative cycle groups.
