We examine Riguet congruences and generalized congruences on a category, giving particular attention to their interrelations from both lattice-theoretic and category-theoretic perspectives. This investigation constitutes the principal contribution of the paper. As an application of these results, starting from a category associated with the free monoid on a set $A$ of sorts, we obtain a skeletal category via the quotient of that category by a suitable Riguet congruence on it. Moreover, we prove that this quotient category is equivalent to the category of finite $A$-sorted sets, while being neither a subcategory of it nor identical to the category of finite $A$-sorted cardinal numbers.
Categories: math.CT