Galactose (Gal) transferase involved in the synthesis of terminal N-acetyllactosamine (LacNac) unit present on glycan chains of glycoproteins and glycosphingolipids. 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. Transfers Gal residue via a beta1->4 linkage to terminal 6-O-sulfoGlcNAc within the LacNac unit of core 2 O-glycans forming 6-sulfo-sialyl-Lewis X (sLex). May contribute to the generation of sLex epitope on mucin-type glycoproteins that serve as ligands for SELL/L-selectin, a major regulator of leukocyte migration. In the biosynthesis pathway of neolacto-series glycosphingolipids, transfers Gal residue via a beta1->4 linkage to terminal GlcNAc of a lactotriaosylceramide (Lc3Cer) acceptor to form a neolactotetraosylceramide. 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 3-O-{beta-D-galactosyl-(1->3)-[6-O-sulfo-N-acetyl-beta-D-glucosaminyl-(1->6)]-N-acetyl-alpha-D-galactosaminyl}-L-seryl-[protein] + UDP-alpha-D-galactose = 3-O-{beta-D-galactosyl-(1->3)-[beta-D-galactosyl-(1->4)-6-O-sulfo-N-acetyl-beta-D-glucosaminyl-(1->6)]-N-acetyl-alpha-D-galactosaminyl}-L-seryl-[protein] + H(+) + UDP 3-O-{beta-D-galactosyl-(1->3)-[6-O-sulfo-N-acetyl-beta-D-glucosaminyl-(1->6)]-N-acetyl-alpha-D-galactosaminyl}-L-threonyl-[protein] + UDP-alpha-D-galactose = 3-O-{beta-D-galactosyl-(1->3)-[beta-D-galactosyl-(1->4)-6-O-sulfo-N-acetyl-beta-D-glucosaminyl-(1->6)]-N-acetyl-alpha-D-galactosaminyl}-L-threonyl-[protein] + H(+) + UDP Protein modification; protein glycosylation. Glycolipid biosynthesis. Interacts with SLC35A2/UGT1. Belongs to the glycosyltransferase 7 family.