I had trouble making sense of the definition of geometric realization of a simplicial set. Let $\Delta^n$ be the standard n-simplex defined as the functor $\hom_\Delta(-,$n$): \Delta \rightarrow$ Set, and $\left|\Delta^{n}\right|$ the topological standard n-simplex, let $X$ be a simplicial set, the realization of $X$ is defined by the colimit:
$\left| X \right| = \underrightarrow{\lim} \: \:\: \left| \Delta^{n} \right|$
$\Delta^{n} \rightarrow X$
in $\Delta\downarrow X$ (the simplex category of $X$).
Frankly I don't understand the notation. Look at the diagram of the colimit in $\Delta\downarrow X$:
$X \cong \underrightarrow{\lim} \: \Delta^{n}$
$\Delta^{n} \rightarrow X$:
Is the geometric realization of $X$ then the geometric realization of the colimit $L = \left| L \right|$? So then $L$ must be standard n-simplex $\Delta^{p}$ for some $p$ no? I want to make sure I'm understanding this right.