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Here we ask if it is possible to have string unification in$\mathsf{SU(5)\times SU(5)}$ gauge group. We specifically investigate theweakly coupled heterotic string unification for the four couplings in thisframework. We show that only a limited versions of $\mathsf{SU(5)\times SU(5)}$with adjoint representation components at intermediated scales, between $M_Z$and $\Lambda$ (unification), are allowed. This is essentially due to thelimitation in the parameter space to the gauge coupling constant$\alpha_1^{-1}$ related to hypercharge. Indeed, only the vanishing hyperchargedecompositions of $\mathsf{SU(5)_L}$ subgroup can help to this unification,namely, the fermion and boson triplets, $\Sigma_3\sim(1,3)_0$, and the fermionand boson octets, $\Sigma_8\sim(8,1)_0$. Thus, these intermediate particles arecompatible with the so-called Adjoint $\mathsf{SU(5)}$. The triplets must livein TeV region and could be accessible at colliders while the octets must alivein very high energy scales. We also show that the non-SUSY unification scenariorequires the introduction of an additional $\mathsf{SU(2)_L}$ scalar colortriplet, $\eta \sim (3,3)_{-{1}/{3}}$, at relatively low energies ($\sim10^{5-11}$GeV), and it may induce the proton decay.

Question for SU(5) x SU(5) string unification