$ of [@Fulton-MacPherson Part II, Theorem 0.0 (0)], that $\alpha$ identifies for a smooth morphism $f: X\to Y$ the orientation $\m O_f:=[\m O_X]\in \bK_{alg}(X \xrightarrow {f} Y)$ with the orientation $\theta(f)\in \bK^{top}_0(X \xrightarrow {f} Y).$  