  A recent signal of 750 GeV diphotons at the LHC can be explained within the
framework of supersymmetric unification by the introduction of vector quarks
and leptons with Yukawa couplings to a singlet S that describes the 750 GeV
resonance. We study the most general set of theories that allow successful
gauge coupling unification, and find that these Yukawa couplings are severely
constrained by renormalization group behavior: they are independent of
ultraviolet physics and flow to values at the TeV scale that we calculate
precisely. As a consequence the vector quarks and leptons must be light;
typically in the region of 375 GeV to 700 GeV, and in certain cases up to 1
TeV. The 750 GeV resonance may have a width less than the experimental
resolution; alternatively, with the mass splitting between scalar and
pseudoscalar components of S arising from one-loop diagrams involving vector
fermions, we compute an apparent width of 10s of GeV.
