For each first-order theory $T$ there is an associated [weak syntactic category](https://www.math.ias.edu/~lurie/278xnotes/Lecture2-Syntax.pdf), sometimes also called "the category of definable sets of $T$" and denoted $\mathrm{Def}(T)$.


Also, for each theory $T$ there is an associated theory $T^\mathrm{eq}$ which can be obtained from $T$ by adding quotient sorts for each definable equivalence relation, see e.g. [here 2.3](http://math.univ-lyon1.fr/~jimenez/Topics_in_Stability_Theory.pdf).


The nLab [claims](https://ncatlab.org/nlab/show/elimination+of+imaginaries#relation_to_indrepresentable_power_objects) that $\mathrm{Def}(T^\mathrm{eq})$ has coproducts.


**Question:** Why? Intuitively, coproducts correspond to the existence of *sum types*. But in $T^\mathrm{eq}$ we only added quotient types.



Why does the category of definable sets of $T^\mathrm{eq}$ have coproducts?
In general, a theory is called proper if every sort is nonempty (in every model) and there is a sort which has (in every model) at least two elements. Then we have the following theorem: if $\mathbb{T}$ is any proper theory, the syntactic category of $\mathbb{T}^{eq}$ is a pretopos. This is a consequence of the following (non trivial) theorem, which is a generalization of the result found in [1]: call a coherent category proper if every object is a subobject of a nonempty one and if there is a decidable object $D$ such that $D$ and $\neg D$ are nonempty, where $\neg D$ is the complement of the diagonal $\Delta: D \to D \times D$. Then the theorem says that any proper coherent exact category is a pretopos.


In the case of a classical first-order theory, whose models have at least two elements, the syntactic category of $\mathbb{T}^{eq}$ is an exact coherent category, and it will be proper, since $\mathbb{T}$ is. Hence, it is a pretopos and thus it has disjoint coproducts. In fact, for a proper coherent category, the exact completion and the coproduct completion are equivalent.


[1] Victor Harnik: "Model theory vs. categorical logic: two approaches to pretopos completion". In Bradd Hart et al., editor, Models, logics, and higher-dimensional categories: a tribute to the work of Mihaly Makkai, volume 53 of CRM Proceedings and Lecture Notes. American Mathematical Society, Providence, R.I., 2011.


