Ideals are intimately related to quotients and congruence relations. They clearly play a very important role in ring theory and order theory. So do normal subgroups in group theory. (Enriched) category theory could be regarded as a common generalization of all these settings. Why is it that such important structures don't work well for categories?

I am aware that there is a categorical notion of congruence relation. However, this doesn't seem to take the spirit of multiple objects to heart: all it does is keep the same objects and relate morphisms within homsets. For one thing, the accompanying notion of quotient category doesn't correspond to coequalizers in $\mathbf{Cat}$ (of which there are many more).

It is not even clear how to define an ideal of a category. To allow for proper ideals, it probably shouldn't simply be a subcategory. Naively one thinks of a subset $I(X,Y)$ of each $\mathrm{Hom}(X,Y)$ that is invariant under composition with arbitrary morphisms, or just of subsets $I(X)$ of each $\mathrm{Hom}(X,X)$, or of $I(X)$ just for some objects; but this doesn't really take objects into account. Thinking of an appropriate definition is even more perplexing for higher categories.

Question: are there related notions of ideal and quotient for categories that have interesting consequences but are not trivial on the level of objects?

It is left open what roles left (postcomposition) or right (precomposition) ideals should play; a related question is if there is a notion of commutativity for categories with interesting consequences.

A convincing explanation why one shouldn't consider such questions would also be a good answer.

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