A sketch is a small category with specified limits and colimits.
A model of a sketch is a Set-valued functor preserving the specified limits and colimits.
A category is called sketchable if it is the category of models of a sketch.
The categories of models of sketches are precisely the accessible categories.
A limit-sketch is a sketch with just limits and no colimits specified.
The categories of models of limit-sketches are the locally presentable categories.
An overview of the theory is given in
An extensive treatment of the links between theories, sketches and models can be found in
Makkai, Paré, Accessible categories: The foundations of categorical model theory Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.1989.
Michael Barr and Charles Wells, Toposes, Triples and Theories?, Originally published by: Springer-Verlag, New York, 1985, republished in: Reprints? in Theory and Applications of Categories, No. 12 (2005) pp. 1-287
That not only every sketchable category is accessible but that conversely every accessible category is sketchable is due to