Is a category one which contains only two objects and **one** morphism between them? Maybe it is a trivial question, but I'm having a hard time in truly understanding this answer, so I believe I need to clarify to myself some probably obvious things first.
Given this description

*

*it contains two objects

*

*$a$

*$b$



*it contains 3 morphisms

*

*$id_a: a \to a$

*$id_b: b \to b$

*$f: a \to b$
Can it represent a category?
I can see the following

*

*it has objects

*it has morphisms

*it has identity morphisms for each object

*composition of morphims are morphisms

So I'd say it is.
 A: Note that composition is part of the data of a category. So in this case we are given a graph consisting of two vertices $A,B$ and three conveniently labeled directed edges $1_A: A \rightarrow A$, $f:A\rightarrow B$, $1_B:B\rightarrow B$. We define composition to be that
$$\begin{align*}
1_A \circ 1_A &= 1_A\\
f \circ 1_A &= f\\
1_B \circ f &= f\\
1_B \circ 1_B &= 1_B
\end{align*}$$
From here on it is straightforward to check that this data indeed satisfies all the axioms of a category (associativity and identity).
This category is important, because it is the free category with an arrow in the following sense. Given any category $\mathcal{C}$ an arrow $\widetilde{f}: \widetilde{A} \rightarrow \widetilde{B}$ in $\mathcal{C}$ induces a unique functor
$$\begin{array}{rcl}
\{A \xrightarrow{f} B\} & \longrightarrow & \mathcal C\\
A & \longmapsto & \widetilde A\\
B & \longmapsto & \widetilde B\\
f & \longmapsto & \widetilde f
\end{array}$$
Conversely any functor $\{A \xrightarrow{f} B\} \longrightarrow \mathcal{C}$ defines a unique morphism $\widetilde f: \widetilde A \rightarrow \widetilde B$ in $\mathcal{C}$ as the image of $f$. These assignments are mutually inverse, so we can deduce that there is a canonical bijection
$$\operatorname{Fun}(\{A\xrightarrow{f}B\},\mathcal{C}) \cong \operatorname{Mor}(\mathcal{C})$$
This makes $\operatorname{Mor}: \mathsf{Cat} \rightarrow \mathsf{Set}$ a representable functor, a concept, which you will either already know or learn soon and which is of utter most importance for category theory.
