In this paper, the direct sum of a finite family of matroids is generalized to that of a finite family of (L, M)-fuzzy matroids. An (L, M)-fuzzy direct sum matroid can be characterized by means of its level L-direct sum matroids. We establish connections between an (L, M)-fuzzy direct sum matroid and its factor (L, M)-fuzzy matroids. The functors preserve the direct sums that exist in both the category of bi-fuzzy matroids and that of fuzzifying matroids, the category of fuzzifying matroids and that of [0, 1]-matroids, the category of [0, 1]-matroids and that of matroids. Finally, we examine the additivity of rank functions for M-fuzzifying matroids and L-matroids.