Abstract: In this paper, it gives the definition of the category of L-sets and bi-induced maps whose true value set is a Locale, a complete Heyting algebra. In this category it defines the L_b-monomorphisms and the L_b-epimorphisms. Especially, it gives the definition, the judgmental theorem of L_b-coequalizers. Furthermore, it defines L_b-regular epimorphisms and proves the judgmental theorem. At the end it concludes a result: the category of L-sets and bi-induced maps is finitely cocomplete. 

Key words: bi-induced map, L_b-monomorphism, L_b-epimorphism, L_b-coequalizer, L_b-regular epimorphism