# Sketches for categories of models of complete theories

In Accessible categories : the foundations of categorical model theory, chapter 3 p.58, Makkai and Paré claim that there is "an (obvious) identification of a class of sketches so that the categories Mod(S) for such sketches S are precisely the categories of models of complete theories with elementary embeddings as morphisms".

In other terms, there seems to be a "sketchable" counterpart of the property of completeness of a formal theory. But there is no explicit reference in the book.

What can be this sketch-theoretical property for such identification? Is there any paper where this identification is explicit?

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The first reference, Theorem 5.42 and Theorem 5.44. These theorems together show that a category is accessible if and only if it is equivalent to a cateory whose objects are models of some theory $T \subseteq L_{\kappa,\kappa}$ and whose morphisms consist of all embeeddings which preserve all formulas of $L_{\kappa, \kappa}$.
In particular this implies that for any first order theory $T$ the category whose objects are of models of $T$ and whose morphisms are elementary embeddings is accessible.