A note on the categorical notions of normal subobject and of equivalence class

In a non-pointed category E, a subobject which is normal to an equivalence relation is not necessarily an equivalence class. We elaborate this categorical distinction, with a special attention to the Mal’tsev context. Moreover, we introduce the notion of fibrant equipment, and we use it to establish some conditions ensuring the uniqueness of an equivalence relation to which a given subobject is normal, and to give a description of such a relation.