Becker and Gottlieb give two constructions of their transfer: a geometric construction using the Pontryagin-Thom map and a more general homotopical construction using Atiyah duality. It is difficult to see that either construction is functorial (even at the level of the homotopy category), and the proof that the first construction is homotopic to the second appears to involve some non-functorial choices. In recent years there has been renewed interest in the problem of both the functoriality and the uniqueness of these constructions. In this talk, we propose axioms for functorial Becker-Gottlieb transfers (which amount to a homotopy coherent version of the Becker-Schultz axioms). We then use our universal characteriziation of Waldhausen K-theory to show that such "fully functorial" transfer maps exist, but they are only unique up to an ambiguity given by the smooth Whitehead space of a point.