What is an 'elementary, pure language'?

Question

In "Category Theory, A Gentle Introduction", by Peter Smith, one can find the following sentence: "Take $\mathcal{L}$ to be the elementary pure language of categories" - what is a 'language', not to mention a 'elementary pure' one?

Answer 1

An elementary language is another term for a first-order theory. The word "elementary" used in this way is somewhat archaic, but it lives on in terms like elementary embedding (an embedding of one first-order theory into another), elementary topos (a first-order theory whose models are a certain kind of category) the Elementary Theory of the Category of Sets (a first-order theory roughly equivalent to ZFC), and elementary class (a term for the class of all models of a fixed first-order theory).

The author uses the phrase "pure sets" to mean that the basic objects of the theory are sets. By analogy, then, a "pure language of categories" means a theory where the basic objects are categories. The author goes on to describe exactly what he's talking about with two basic sorts of objects and morphisms. There are a few unary and binary operations added in, and some equalities enforced. This signature is precisely what's meant by an elementary language (first-order theory).