  Studies of the Deep Inelastic Scattering (DIS) have provided fundamental
information of the nucleon structure for decades. The electron-ion collider
(EIC) will be the first collider capable of DIS study with both polarized
lepton and polarized hadron beams, providing the possibility of accessing new
electroweak structure functions of the nucleon. In this work, we completed the
DIS cross section derivations for both longitudinally and transversely
polarized leptons and hadrons, with no approximations made, and with all three
contributions $\gamma \gamma, \gamma Z, ZZ$ included. These results were
derived using primarily tensor algebra and Feynman calculus, starting from
previously established leptonic and hadronic tensors and carry out their
contraction. Our results are presented in terms of both spin-averaged and
spin-dependent cross sections, allowing direct comparison with experimentally
measured cross sections and their asymmetries. We include also in our
discussion comparisons of different conventions that exist in the literature.
