I am confused about some notation in the quick way of constructing the geometric realization of a simplicial set.
Consider the simplex category $\Delta \downarrow X$ of a simplicial set $X$.
The objects are the maps (natural transformations) $\sigma:\Delta ^n \to X$ where $\Delta^n= \Delta(-, [n]): \Delta^{op} \to \textbf{Set}$ is the simplicial set.
An arrow between $\sigma: \Delta^n \to X$ to $\tau: \Delta^m \to X$ is the map (natural transformation) $\theta : \Delta^n \to \Delta^m$ induced by the map $\theta*: [m] \to [n]$ in $\Delta^{op}$ such that the following holds:
$$\tau \circ \theta = \sigma$$
The Geometric Realization of a simplicial set $X$ is defined as:
$$|X| = \lim_{\rightarrow \\ {\Delta^n\to X} \\ in \Delta\downarrow X}|\Delta^n|$$
I am confused about the notation here about what the diagram $F$ of this colimit $\lim F$ is exactly.
The diagram $F:\Delta\downarrow X\to \mathrm{Top}$ sends $(\sigma:\Delta^n\to X)$ to $|\Delta^n|$ and sends a map in $\Delta\downarrow X$ induced by $\theta:\Delta^m\to \Delta^n$ to $|\theta|$. In other words, there is a natural projection functor $p:\Delta\downarrow X\to \Delta$, and $F$ is the composition of $p$ with the canonical functor $\Delta\to\mathrm{Top}$.