Factoring the Becker-Gottlieb transfer through the trace map

In 1998, Becker and Schultz [6] published axioms characterizing the Becker-Gottlieb transfer τBG (p):E^∞(B₊) → E^∞ (E₊) for certain types of fibrations p: E → B. We verify these axioms for the composite of the algebraic K-theory transfer τK (p) E^∞ (B₊) → A(E) of any perfect fibration p followed by the evaluation (at the unit) from the free loop space Λ of the Bökstedt trace map tr: A(E) → E^∞ (ΛE₊) → E^∞ (E₊). As a consequence, for p any compact ANR fibration with finite CW base (those considered by Becker-Shultz), τBG (p) tr τK (p).