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A multi-object spectrograph is a spectrograph designed to collect spectrographic data from multiple targets simultaneously. A spectrograph open to the image (i.e., no restricting slit) can do this (producing a little "rainbow" for each imaged star), but the term is generally used for newer variations.
Generally, it uses a method to bring in light from known locations within the telescope's focal plane image, and requires preparation in terms of adjusting the instrument for these specific locations. This has been done by using a mask with multiple slits rather than a single slit.
It has also been done using what amounts to an imaging spectrograph or integral field spectrograph but designed so the "pixels" can be placed so as to gather data from specific target locations rather than being spread evenly over a field. An integral field unit of the "fiber" type might have the fibers rearranged to certain points in the focal plane image.
Telescopes/instruments using this:
The term multi-object spectrography indicates observation with such instruments.