A category object in an $\infty$-category

Question

I am quite confused on what a category object in an $(\infty,1)$ category here.

How did we get the diagram

$$X(\{0,1\}) \times \cdots \times X(\{n-1,n\})$$ in the first place ?

If we were to spell out the definition of an $(\infty, 1)$-limit, we would want a simplicial set $K$, and a morphism $p:K \rightarrow C$ that gives the diagram written above.

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I am guessing the definition should be that we take $K$ as the nerve of $$ \{0 \} \rightarrow \{0,1\} \leftarrow \cdots \rightarrow \{n-1,n\} \leftarrow \{n \}$$ which maps in to $\Delta$ by sending one element set to $[0]$ and two element sets to $[1]$. Then there is a canonical $K \rightarrow K^{\triangleleft} \rightarrow C$ that extends our diagram via the sending the cone point to $[n]$.

Answer 1

You can find the diagram in definition 2.2 here.