*2.1. First, Some Category Theory*

Category theory of Eilenberg and Mac Lane [19,54] constitutes the mathematical substance of institution theory. This situation is similar in other axiomatic approaches to model theory, such as in the above-mentioned Budapest school of abstract model theory. This means that the mathematical structures in institution theory are all categorical. On the other hand, the flow of ideas in institution theory is model theoretic. So, institution theory is a form of model theory that at the level of the mathematical structures is heavily based on categorical structures. This represents a sharp contrast to the widespread perception of category theory as a mere language that supports a clearer presentation and structuring of mathematical concepts that in fact do not have an inherent categorical nature. Institution theory without category theory is possible to the same extent as, for instance, group theory is possible without set theory!

Why such a reliance on category theory; is it indispensable for the axiomatic treatment of model theory? There are several reasons for this. One is that that set theoretical structures cannot support the required level of generality and abstraction. Another one is that category theory emphasises the relationships between objects rather than their internal structures. Moreover, category theory is conceptually a highly developed area of mathematics, so this brings in much conceptual and technical power.

However, the level of category theory involved in institution theory is rather elementary, as it hardly touches advanced concepts and techniques; the only slight exception being found in the area of stratified institutions. So, familiarity with concepts such as opposite (dual) of a category **C** (denoted **C**-), comma category, functor, (lax) natural transformation, (co)limit, and adjunction may be enough to be able to engage with the study of institutional model theory. In this survey, with a few exceptions, in general, we follow the terminology and the notations of [19]. As regards the notational conventions,

