Change of Base Using Arrow Categories

Arrow categories establish a suitable framework to reason about L-fuzzy relation abstractly. For each arrow category we can identify the Heyting algebra L that is used as the lattice of membership or truth values by the relations of the category. Therefore, arrow categories model the fixed-base approach to L-fuzziness, i.e., all relations of the given arrow category use the same membership values. In this paper we are interested in the process of changing the base, i.e., an operation that allows to switch from an L-1-fuzzy relation to an L-2-fuzzy relation by replacing all membership values from L-1 by values from L-2. We will define and investigate this change of base between two abstract arrow categories for which component-wise reasoning cannot be performed.