Let $[n]$ be the category of integers $\{0, \ldots, n \}$ totally ordered by $\le$ (its morphisms). Let $X:[n] \rightarrow C$ be a functor to another category $C$. Then
is $X(n)$ a colimit of the diagram given by $X$?
This seems to be deceptively true:
$X(n)$ is a cocone, with $id:X(n) \rightarrow X(n)$ be the leg at $n$.
If $Y$ is a cocone then there is a morphism $y_n:X(n) \rightarrow Y$. This is our morphism from the cocones too.
This is unique as any such cocone morphism $y:X(n) \rightarrow Y$ must satisfy, $y \circ id= y_n$.