Operating on 6 interferometric baselines, i.e. using all 4 unit telescopes (UTs) of the Very Large Telescope Interferometer (VLTI) simultaneously, the 2nd generation VLTI instrument GRAVITY will deliver narrow-angle astrometry with 10μas accuracy at the infrared K-band. At this angular resolution, GRAVITY will e.g. be able to detect the positional shift of the photo-center of a flare at the Galactic Center within its orbital timescale of about 20 minutes, using the observed motion of the flares as dynamical probes of the gravitational field around the supermassive black hole Sgr A*. Within the international GRAVITY consortium, the 1. Physikalische Institut of the University of Cologne is responsible for the development and construction of the two spectrometers of the camera system: one for the science object, and one for the fringe tracking object, both being operated in cryo-vacuum conditions. In this contribution we describe the basic functionality of the two units and present the final optical design of the two spectrometers as it got realised successfully until end of 2013 with minor changes to the Final Design Review (FDR) of October 2011. In addition we present some of the first light images of the two spectrometers, taken at the laboratory of the Cologne institute between Dec. 2012 and Oct. 2013 respectively. By the end of 2013 both spectrometers got transferred to the PI institute of GRAVITY, the Max-Planck-Institute for Extraterrestrial Physics, where at the time of writing they are undergoing system-level testing in combination with the other sub-systems of GRAVITY.