Elementary classes in the injective subcategories approach to abstract model theory (Q1094414)

In model theory it is well known that a class is elementary if and only if it is closed under elementary equivalence and ultraproducts. In a constructive way we can say that given a language L and a class K of L- structures, the models of the underlying theory are L-structures elementarily equivalent to an ultraproduct of objects of K. The author transposes this result to the categorical language of Andréka, Németi, etc. The principal problems are those of definition; the categorical conditions necessary for this translation to remain valid are reasonable.