  Let $Covering$ be the category of the category of fuzzy coverings, and
$Partition$, the category of fuzzy partitions. We geometrically construct an
isomorphism of categories between $Partition$ and a full subcategory of
$Covering$, which can be used to derive bijections between fuzzy partitions and
fuzzy coverings with finitely many sets. Also, we establish an isomorphism
between $Covering[n]$, the category of coverings with $n$ fuzzy sets, and a
subcategory of $Partition$, whose objects are partitions with $n$ sets which
satisfy certain conditions, which can be also used to deduce another bijection
between fuzzy partitions and fuzzy coverings with finitely many sets.
