# Notation for Geometric realization of simplicial sets

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 $$\Delta\to\mathrm{Top}$$.