The aim of this paper is to construct an L-valued category whose objects are L-E-ordered sets. To reach the goal, first, we construct a category whose objects are L-E-ordered sets and morphisms are order-preserving mappings (in a fuzzy sense). For the morphisms of the category we define the degree to which each morphism is an order-preserving mapping and as a result we obtain an L-valued category. Further we investigate the properties of this category, namely, we observe some special objects, special morphisms and special constructions.
In this paper, we introduce the product, coproduct, equalizer and coequalizer notions on the category of fuzzy implications on a bounded lattice that results in the existence of the limit, pullback, colimit and pushout. Also isomorphism, monic and epic are introduced in this category. Then a subcategory of this category, called the skeleton, is studied. Where none of any two fuzzy implications are Φ-conjugate.