Is there a precise definition to "interpretation" in Mathematics?

Question

A category (as distinguished from a metacategory) will mean any interpretation of the category axioms within set theory. Here are the details.

Sect-2,page-10, Category Theory for the Working Mathematician.

Is there an exact definition of interpretation in Mathematics? Also is there a general meta theory about interpretable structures?

Related

Answer 1

Per nLab,

In Categories for the Working Mathematician, Saunders Mac Lane uses the term ‘metacategory’ to mean any model of the first-order theory of categories, and reserves the word ‘category’ for a metacategory whose objects and morphisms form sets.

A category in general doesn't have to be representable by a set (a "set" depending on the foundation of mathematics you are using), in fact such categories are generally called "small". You can see in this question some examples of "large" categories.

I don't believe interpretation has any specific meaning in mathematics, it's just the word Mac Lane uses here to describe the idea of restricting the axioms only to sets.