Definition: A category is a class $\mathcal{C}$ of objects (denoted $A, B, C, ...$) together with
(i) a class of disjoint sets, denoted $\text{hom}(A,B)$, one for each pair of objects in $\mathcal{C}$; (an element $f$ of $\text{hom}(A,B)$ is called a morphism from $A$ to $B$ and is denoted $f:A\to B$);
(ii) for each triple $(A,B,C)$ of objects of $\mathcal{C}$ a function $$\circ: \text{hom}(B,C)\times \text{hom}(A,B)\to \text{hom}(A,C)$$ (for morphisms $f:A\to B$, $g:B\to C$, this function is written $(g,f)\mapsto g\circ f$ and $g\circ f: A\to C$ is called the composite of $f$ and $g$); all subject to the two axioms:
(I) Associativity. If $f:A\to B$ $g:B\to C$, $h:C\to D$ are morphisms of $\mathcal{C}$, then $h\circ (g\circ f)=(h\circ g)\circ f$.
(II) Identity. For each object $B$ of $\mathcal{C}$ there exists a morphism $1_B:B\to B$ such that for any $f:A\to B$, $g:B\to C$, $$1_B\circ f=f \text{ and } g\circ 1_B=g.$$
From this post, above definition is of locally small category, because $\text{hom}(A,B)$ is a set.
Question: This is my first time seeing definition which don’t feel rigioursly precise. I mean words object and morphism in the definition are not defined at all. I assume object is inherently difficult to define. In naive set theory, set is defined as collection of objects. Do we face similar difficulty in defining object in category as in set?
How morphisms are defined? Set of all morphism from $A$ to $B$ is $\text{hom}(A,B)$ or element of $\text{hom}(A,B)$ is called morphism from $A$ to $B$? That is, which came first $\text{hom}(A,B)$ or morphism from $A$ to $B$.
I think author assumed $\text{hom}(A,B)$ to be set for all $A,B\in \text{ob}(\mathcal{C})$ because in the book there is no definition (or at least not much discussion) of cartesian product of classes and function on classes. Without these notion composition $\circ$ do not make sense.
(i) is saying $\{\text{hom}(A,B)\mid A,B\in \text{ob}(\mathcal{C})\}$ is a class and $\text{hom}(A,B) \cap \text{hom}(C,D)=\emptyset$ if $(A,B)\neq (C,D)$. Am I right?
Edit: I find wikipedia definition of category extremely precise and unambiguous. Category theory and Category (mathematics).
"Object" and "morphism" are simply words. The class $\mathcal{C}$ is just a class, containing elements. What those elements "are" is irrelevant. What is the number $1$ made of? We just are calling elements of $\mathcal{C}$ "objects" (of the category) because that is far nicer and more intuitive than just stating 'elements of $\mathcal{C}$' every time.
$\mathrm{hom}(A,B)$ is defined, by your author to be a set. $(i)$ is correct.
Similarly "morphism" is simply a word for an element of a set $\mathrm{hom}(A,B)$. What these elements fundamentally "are" is again irrelevant. The author could have stated this definition without using the words 'object' or 'morphism' just fine.