Categories and sorted systems Maclane gives a definition (in a survey on categorical algebra) of category in the following way (in my own words), a category is a two-sorted system the sorts being called the objects and the arrows and that satisfies certain properties...
My questions are 
What is this system, is a logic, a theory?
Why a category is a two-sorted system?
Then, where are the categories, in another system?
 A: Maclane is saying that category theory can be formalized in a 2-sorted first-order logic. See https://en.wikipedia.org/wiki/First-order_logic#Many-sorted_logic for the basic definition & more. In many-sorted logic you can have multiple discrete domains of individuals; each constant is of a particular sort, and the signature of each argument to a predicate or function includes requirements about which sort(s) of individual can "go there".
Many-sorted first-order logic is really just a notational convenience, it doesn't extend the power of first-order logic[1]. You can eliminate the multiple "sorts" using a predicate for each sort, adding axioms that everything is of some sort and nothing is of more than one sort; the other axioms of the theory have to be modified using these predicates to limit scope to things of the intended sort(s) as needed.
[1]: This paragraph is true when there are finitely many sorts. With infinitely many sorts, however, you can't say "everything is of some sort", as formulas are finite, though with infinitely many sentences you can say that nothing is of any two distinct sorts. See Alex Kruckman's comment for another difference when there are infinitely many sorts.
