I have read yesterday the following statement:
An initial object when it exists is the colimit of the empty diagram. A terminal object when it exists is the limit of the empty diagram.
And I found some justification of the second statement here:
What means an empty diagram? but still the justification is not very clear to me, I am wondering if someone can explain this to me in a simpler way please?
A precise way to understand this is as follows :
Note that for a diagram $D : I \rightarrow C$, its colimit, if it exists, represents the functor $Cocone(D,-) : C \rightarrow Set$ which maps an object $c \in C$ onto the set of cocones on $D$ with vertex $c$. In other words, $$Cocone(D,c) \cong C(colim\text{ }D, c)$$ naturally in $c \in C$.
In the specific case that the shape category $I = \emptyset$, $Cocone(D,c)$ (which is the set of all natural transformations $D \rightarrow \Delta_c$) is a singleton set (namely, the natural transformation with no components).
Thus, for any object $c$, $C(colim\text{ }D, c) \cong Cocone(D,c)$ is a singleton set. Thus, $Colim\text{ }D$ is an initial object.