In this paper, we analyze the high-resolution UV spectra for a C1.7 solar flare (SOL2017-09-09T06:51) observed by the textit{Interface Region Imaging Spectrograph} (textit{IRIS}). {We focus on the spectroscopic observations at the locations where the cool lines of ion{Si}{4} 1402.8 AA ($sim$10$^{4.8}$ K) and ion{C}{2} 1334.5/1335.7 AA ($sim$10$^{4.4}$ K) reveal significant redshifts with Doppler velocities up to $sim$150 km s$^{-1}$.} These redshifts appear in the rise phase of the flare, then increase rapidly, reach the maximum in a few minutes, and proceed into the decay phase. Combining the images from textit{IRIS} and Atmospheric Imaging Assembly (AIA) on board the {em Solar Dynamics Observatory} ({em SDO}), we propose that the redshifts in the cool lines are caused by the downflows in the transition region and upper chromospheric layers, which likely result from a magnetic reconnection leading to the flare. In addition, the cool ion{Si}{4} and ion{C}{2} lines show gentle redshifts (a few tens of km s$^{-1}$) at some other locations, which manifest some distinct features from the above locations. This is supposed to originate from a different physical process.
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