I am reading the famous papaer Classifying spaces and spectral sequences by Segal and I am a little confused by something.
I am familiar with Milnor's join construction of classifying spaces. Let us denote by $\mathbb E G \to \mathbb B G $ the universal map one obtains this way.
I also know that there is a way to build a universal $G$-principal bundle via simplicial sets, namely, one takes the base to be $BG:=\lvert N(G) \rvert$, the realization of the nerve of $G$, and the total space is the realization of a further simplicial set whose definition I will not recall here.
I know by abstract formalism that the things are homotopy equivalent, and this is fine.
But now I would like to focus on explicit identifications. Segal defines, given a small category $C$, the category unravelled over $\mathbb N$; denoted by $C_{\mathbb N}$ and he claims that the classifying space obtained via the Milnor join construction satisfies $$ \mathbb B G = \lvert N(G_{\mathbb N}) \rvert.$$
Why? Is this obvious? The way he writes down, I expect this to be seen by inspection just by looking at the definitions, but I can not see this.