A category C is composed of

• a collection Ob(C) of objects: A, B, C, ...
• a collection Ar(C) of Morphisms/arrows
• Mappings $dom, cod: Ar(C) \to Ob(C)$, specifying for each morphism which object is its domain, and which is its codomain. An arrow $f$ from $A$ to $B$ is written $f: A \to B$. For each such pair we define the set $C(A,B) = \{f \in Ar(C) | f: A \to B\}$. We refer to this set as hom-set. Note that distinct hom-sets are disjoint
• For any triple of objects, a composition map $c_{A,B,C} : C(A,B) \times C(B,C) \to C(A,C)$. $c_{A,B,C}(f,g)$ is written $g \circ f$ or sometimes $f;g$ (as often it refers to Function composition. Diagramatically, $A\to_f B \to_g C$.
• For each object $A$, an identity arrow $id_A: A \to A$.

The composition map must satisfy Associativity (in the same way as Function composition). Also $f \circ id_A = f = id_A \circ f$. whenever the domains and codomains of the arrows match appropriately so that the compositions are well-defined.

One can naturally define a Subcategory

things which are isomorphic in a category tend to be undistinguishable that is if a property applies to one, it applies to the other too