We have used Far Ultraviolet Spectroscopic Explorer (FUSE) to obtain a series of 13 observations of the nearby dwarf nova VW Hyi that cover the period from the end of a superoutburst through the following normal outburst of the system. Here, we present the quiescent spectra taken after each outburst event. The spectra obtained during quiescence contain at least three components. The dominant component over most of the FUSE wavelength range is the white dwarf (WD), which cools following the superoutburst. The amount of cooling is dependent on the WD models utilized. For log g of 8.0, the temperature drops from 24,000 K just after the outburst to 20,000 K just before the normal outburst. For this model, and for a distance of 65 pc, the radius of the WD is approximately 8 x 10(8) cm and nu sin (i) is similar to 420 km s(-1). The fact that the derived radius is smaller than expected for a WD with log g = 8 suggests a higher gravity WD or that VW Hyi is somewhat further than its canonical distance of 65 pc. Either is possible given the current uncertainty (+/- 20 pc) in the distance to VW Hyi. Earlier suggestions that the WD photosphere show evidence of CNO processed material are confirmed, but our analysis also highlights the fact that significant issues remain in terms of analyzing the spectra of WDs in such unusual physical situations. The second component is relatively featureless and shows substantial modulation on the orbital (and just after outburst, the superhump) period. The second component is most likely associated with the hot spot where material from the secondary encounters the disk, rather than emission from the boundary layer region between the inner disk and WD. This second component fades about 10 days after the superoutburst. There is also a third component, clearly visible in terms of broad emission lines of C III lambda 977, N III lambda 991, and a combination of Ly beta and O VI.1032, 1038, which appears to be accompanied by a flat continuum. The strength of the emission lines, which are almost surely associated with the accretion disk, appears relatively constant for the duration of the observations.