Are a categories objects unique? The following commutative diagram comes from Awodey's Category theory (p. 4):

Are objects allowed to appear more than once in a commutative diagram describing a category? From what I understand, the nodes and arrows represent objects and morphisms respectively. Does that mean this category has $4$ objects, two copies of $A$ and two copies of $B$?
 A: There is only one object $A$ and one object $B$. Yes, they are depicted twice in the diagram, but this does not mean that we have four distinct objects.
You can regard a commutative diagram as a collection of equations involving certain morphisms of the given category $\mathcal C$. Awodey's diagram does not use any individual properties of $\mathcal C$ or of the morphisms occurring in the diagram. Actually the diagram is completely tautological: The left triangle says that $f \circ 1_A = f \circ 1_A$ and the right triangle $1_B \circ f = 1_B \circ f$. A slightly more interesting diagram is obtained by replacing the "diagonal" arrows by $f$. This diagram says that  $f \circ 1_A = f$ and $1_B \circ f = f$, i.e. graphically depicts the defining property of identity morphisms. These two equations do not involve four distinct objects and five distinct morphisms, but only the two objects $A, B$ and the three morphisms $f, 1_A, 1_B$ between these two objects.
As a very simple analogue let us consider the equation
$$ 1 + 2 - 1 = 2 . \tag{1}$$
The integers $1, 2$ occur twice, but this does not mean that $(1)$ involves four distinct integers. There exist only one integer $1$ and one integer $2$, but both are multiply written in  $(1)$.
Let us now give a formal definition of a diagram.
A diagram in $\mathcal C$ consists of

*

*A directed graph $\mathcal{G}$ with set of vertices $V$ and set of edges $E$ (note that an edge is nothing else than an ordered pair $(v,w)$ of vertices).


*Functions $\phi_V  : V \to Ob(\mathcal C)$ from $V$ to the class of objects of $\mathcal C$ and $\phi_E  : E \to Mor(\mathcal C)$ from $E$ to the class of morphisms of $\mathcal C$ such that $\phi_E(v,w)$ is a morphism $\phi_V(v) \to \phi_V(w)$.
The diagram is commutative if for all finite paths in $\mathcal{G}$ connecting the same vertices $v, w$ the associated compositions of morphisms under $\phi_E$ yield the same morphism $\phi_V(v) \to \phi_V(w)$.
In Awodey's diagram the graph consists of four (distinct!) vertices and five (distinct!) edges. To these structure elements we associate two distinct objects and five distinct morphisms.
