This article is devoted to present the degree to which a mapping defined from
L
X
{L}^{X}
to
M
,
M,
which is an
(
L
,
M
)
\left(L,M)
-fuzzy bornology in the sense of Liang et al. In this respect, the degree to which
L
L
-subset is bounded according to the given bornological structure is described. In addition, the notions of boundedness degree and the boundedness preserving degree of a mapping defined between bornological spaces are introduced. The characterizations of the provided notions are studied.