So we know that geometric realization of simplicial set $X$ is just colimit of this functor F as seen here: Notation for Geometric realization of simplicial sets.
And we can also state the geometric realization as some quotient topology, as seen below.
Since the geometric realization is the colimit, this means that we should be able to find a colimit cone
$\alpha \colon F \rightarrow \left|X\right|$. But what is this colimit cone $\alpha$ explicitly?
That is, how is $\alpha_{\sigma\colon \triangle^n \rightarrow X} \colon \left|\triangle^n\right| \rightarrow \left|X\right|$ defined as a function, where $\left|X\right|$ is given by definition 7 in the image?
Thanks!