Given a category $\mathcal{C}$, we have a nerve functor $$\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$$ that assigns to $\mathcal{C}$ its nerve $\mathrm{N}(\mathcal{C})$. This functor seems to have a left adjoint $$\tau_1 \colon \mathbf{Set}_{\Delta} \to \mathbf{Cat}$$ that assigns to a simplicial set $X$ its fundamental category, as in Joyal's Notes on Quasi-Categories.
There it also states that the fundamental grouped $\pi_1 X$ is obtained by inverting the arrows of $\tau_1 X$, but there is no construction of $\tau_1 X$.
What is the construction/definition of the fundamental category of a simplicial set $\tau_1 X$? What are its objects and morphisms?
The best presentation that I know of is in Riehl and Verity: 1.1.10&11