  Limits and colimits of diagrams, defined by maps between sets, are universal
constructions fundamental in different mathematical domains and key concepts in
theoretical computer science. Its importance in semantic modeling is described
by M. Makkai and R. Par\'e, where it is formally shown that every axiomatizable
theory in classical infinitary logic can be specified using diagrams defined by
maps between sets, and its models are structures characterized by the
commutativity, limit and colimit of those diagrams. Z. Diskin taking a more
practical perspective, presented an algebraic graphic-based framework for data
modeling and database design. The aim of our work is to study the possibility
of extending these algebraic frameworks to the specification of fuzzy
structures and to the description of fuzzy patterns on data. For that purpose,
in this paper we describe a conservative extension for the notions of diagram
commutativity, limit and colimit, when diagrams are constructed using relations
between fuzzy sets, evaluated in a multi-valued logic. These are used to
formalize what we mean by "a relation $R$ is similar to a limit of diagram
$D$," "a similarity relation $S$ is identical to a colimit of diagram $D$
colimit," and "a diagram $D$ is almost commutative."
