In definition of a category , what is the meaning of 'consists of'

Question

A category $\mathsf C$ consists of the following three mathematical entities:

• A class $\operatorname{ob}(\mathsf{C})$, whose elements are called objects;

• A class $\hom(\mathsf{C})$, whose elements are called morphisms or maps or arrows. Each morphism $f$ has a source object $a$ and target object $b$.

• A binary operation $\circ$, called composition of morphisms, such that for any three objects $a$, $b$, and $c$, we have $\hom(b, c) \times \hom(a, b) \to \hom(a, c)$. The composition of $f : a \to b$ and $g : b \to c$ is written as $g \circ f$ or $gf$, governed by two axioms: [...]

What the exact meaning of 'consist of' in the first sentence? Of course, I know the usual meaning. However, since it is not a mathematical term, I don't know the mathematical meaning of 'consists of'.

Answer 1

To say an object "consists of" followed by a list of entities means that entities are the data types you have to specify to describe the object.

"Consists of" in the case of the definition of category means that to define a category, you have to specify three mathematical objects: Two classes and a binary operation, and they must satisfy the requirements in the definition.

The definition using an ordered triple also works, but it has the minor problem that the three entities are specified in order, which adds a superfluous fact that is not necessary to the definition. No one thinks that the "first part" of a category is its class of objects. It is not wrong to define a category as a triple, but it certainly adds unnecessary structure and is therefore inelegant.

Answer 2

To be formal, you can say that a category is a triple $(Ob(C), Hom(C), \circ)$ such that, etc ...

The notion of triple is perfectly and formaly defined in set theory.

Of course, I use the definition of category which states that $Ob(C)$ and $Hom(C)$ must be sets. To work with this definition, one usually uses Grothendieck Universes.