While modern software engineering, with good reason, tries to establish the idea of reusability and the principles of parameterization and loosely coupled components even for the design of performance-critical software, Fast Fourier Transforms (FFTs) tend to be monolithic and of a very low degree of parameterization. The data structures to hold the input and output data, the element type of these data, the algorithm for computing the so-called twiddle factors, the storage model for a given set of twiddle factors, all are unchangeably defined in the so-called butterfly, restricting its reuse almost entirely. This paper shows a way to a component-based FFT by designing a parameterized butterfly. Based on the technique of lifting, this parameterization includes algorithmic and implementation issues without violating the complexity guarantees of an FFT. The paper demonstrates the lifting process for the Gentleman-Sande butterfly, i.e., the butterfly that underlies the large class of decimation-in-frequency (DIF) FFTs, shows the resulting components and summarizes the implementation of a component-based, generic DIF library in C++.