  The notion of a categorical quotient can be generalized since its standard
categorical concept does not recover the expected quotients in certain
categories. We present a more general formulation in the form of
$\mathcal{F}$-quotients in a category $\mathbf{C}$, which are relativized to a
faithful functor $\mathcal{F}\colon \mathbf{C} \to \mathbf{D}$. The isomorphism
theorems of universal algebras generalize to this setting, and we additionally
find important links between $\mathcal{F}$-quotients in the concrete category
of first-order structures, and quotients defined for model-theoretic
equivalence classes. By first working in this categorical setting, some
quotient-related results for first-order structures can be naturally obtained.
In particular, we are able to prove some isomorphism theorems in the context of
model theory directly from their corresponding categorical isomorphism
theorems.
