Consider the sequence of posets $$[0] \to [1] \to [2] \to [3] \to...$$ where $[n]= \{0 \leq 1 \leq \dots \leq n \}$ where the maps $[n] \to [n+1]$ are inclusion maps. Find the limit and colimit of this sequence, viewed as a diagram over the poset category $\mathbb{N}$.
How can the sequence be viewed as a diagram? Indeed, a diagram of type $J$ in $C$ is a functor $D: J \to C$ from an index category $J$ to a category $C$. I think the index category here is simply the category with objects $\mathbb{N}$ and maps as inclusions and the category $C$ is the same. Is this true?
You are correct about your definition of diagram. The $C$ in your case is Poset, the category of partially ordered sets and monotonic functions. Each $[n]$ is a partially ordered set. The index category is the specific poset $\mathbb{N}$ with the usual ordering relation, i.e. $\leq$. This can be viewed as a category whose objects are natural numbers, i.e. elements of $\mathbb{N}$ and for which there exists a (single) arrow $n \to m$ iff $n \leq m$. This category is the $J$ in your definition of diagram. You could describe the object part of the actual diagram as $D(n) = [n]$ with the obvious action on arrows.