In any group $ G $, we can form normal multi-fuzzy subgroups and normal multi-anti fuzzy subgroups. In this paper, we discuss the properties of normal multi-fuzzy subgroups and normal multi-anti fuzzy subgroups. Based on this structure, we construct the factor groups relative to a normal multi-fuzzy subgroup or normal multi-anti fuzzy subgroup. This gave rise to a one-to-one correspondence between the normal multi-anti fuzzy subgroups of a group $ G_2 $ and those of a group $ G_1 $ which are constant on the kernel of homomorphism $ f $. This correspondence can occur if $ f $ is epimorphism of groups from $ G_1 $ to $ G_2 $.