Galactose (Gal) transferase involved in the biosynthesis of glycoproteins, proteoglycans, and glycosyphingolipids. Catalyzes the transfer of Gal residue via a beta1->4 linkage from UDP-Gal to the non-reducing terminal N-acetyl glucosamine 6-O-sulfate (6-O-sulfoGlcNAc) in the linearly growing chain of both N- and O-linked keratan sulfate proteoglycans. Cooperates with B3GNT7 N-acetyl glucosamine transferase and CHST6 and CHST1 sulfotransferases to construct and elongate mono- and disulfated disaccharide units [->3Galbeta1->4(6-sulfoGlcNAcbeta)1->] and [->3(6-sulfoGalbeta)1->4(6-sulfoGlcNAcbeta)1->] within keratan sulfate polymer. N-acetyl-D-glucosamine + UDP-alpha-D-galactose = beta-D-galactosyl-(1->4)-N-acetyl-D-glucosamine + H(+) + UDP a beta-D-GlcNAc-(1->3)-beta-D-Gal-(1->4)-beta-D-Glc-(1<->1)-Cer(d18:1(4E)) + UDP-alpha-D-galactose = a neolactoside nLc4Cer(d18:1(4E)) + H(+) + UDP Protein modification; protein glycosylation. Trans cisternae of Golgi stack. Belongs to the glycosyltransferase 7 family.