Category Theory: Functors as Diagrams I have began classes on Category Theory, I do not understand the concept of 'functors as diagrams' the book I am using is Category Theory and Applications by Marco Grandis.
I have been asked to prove the following statements:

*

*Every functor defined on a preordered set $S$ is automatically a commutative diagram.


*A functor $1 \longrightarrow C$ amounts to an object of $C$ while a functor $2 \longrightarrow C$ amounts to a morphism of C.


*A functor $X : (\mathbb{N},\leq) \to C$ defined on the ordered set of natural numbers (i.e. the ordinal $\omega$), amounts to a sequence of consecutive morphisms of $C$
\begin{equation*}
  X_0 \longrightarrow X_1 \longrightarrow X_2 \longrightarrow \dots \longrightarrow X_n \longrightarrow \dots
 \end{equation*}
while the functor $X : (\mathbb{N},+)\to C$, defined on the additive monoid of natural numbers, amounts to an endomorphism $u_1 : X_* \to X_*$ $(\text{and its powers} \quad u_n=(u_1)^n$, including $u_0=id(X_*))$
 A: The basic idea to understand diagrams as functors is that we are associating some sort of map (the functor) with its image (the visual diagram).
As you can read on wikipedia, a diagram in category $C$ is a covariant functor $D:J \rightarrow C$ where $J$ is called an index category. It doesn't really matter what the objects and arrows in $J$ are. Instead it is the connectivity that is important. $D$ associated objects and arrows in $J$ with those in $C$ so that the relationships are preserved. You might find this blog link with nice images helpful.
In statement 2, $1$ is the category with a single object and a single arrow. What is the possible image of the functor $1\rightarrow C$? Similar thinking is useful to decode $2 \rightarrow C$ where $2$ is the category $\bullet \rightarrow \bullet$.
Build upon this idea in statement 3. What does the category $(\mathbb{N}, \le)$ look like? Then what is the possible image of a functor $X:(\mathbb{N}, \le)\rightarrow C$ look like? For the case involving $(\mathbb{N},+)$, think carefully about how a monoid is understood as a category.
