  We show that the Waldhausen trace map $\mathrm{Tr}_X \colon A(X) \to QX_+$,
which defines a natural splitting map from the algebraic $K$-theory of spaces
to stable homotopy, is natural up to \emph{weak} homotopy with respect to
transfer maps in algebraic $K$-theory and Becker-Gottlieb transfer maps
respectively.
