Let $X \in \mathbf{Set}_{\Delta}$ an $\infty$-category and $\tau_1$ the left adjoint functor to the nerve $\mathrm{N} \colon \mathbf{Cat} \to \mathbf{Set}_{\Delta}$.
Show that if $x$ is an initial object in $X$, then $x$ is an initial object in $\tau_1(X)$.
I don't really know where to start here. An object $x \in X$ is initial if the map $h \colon X_{/x} \to X$ is a trivial Kan fibration (by which I guess it means that it is both a weak homotopy equivalence and a Kan fibration), where $h$ is defined by restricting $$\Delta^0 \star \Delta^n \to \mathcal{C}$$ to $\Delta^n$.
On the other hand, objects in $\tau_1(X)$ are the $0$-simplices of $X_0$. But I don't know how to prove the result from that. Thank you in advance.
Here are some hints. First, you need a much better idea of what a trivial (Kan is often omitted) fibration is. It's a Kan fibration and a weak equivalence, but much more usefully it's a map which permits lifting against all cofibrations. Equivalently, against all generating cofibrations, namely the inclusions $\partial \Delta^n\to \Delta^n$. In short, $x$ is initial when fillings of a sphere under $x$ (that is, a map $\partial\Delta^n\to X_{x/}$) in $X$ lift to fillings in $X_{x/}$.
Now, you should consider what this means for small $n$. Note that the boundary of $\Delta^0$ is empty; so what do we know about an initial $x$ from the $n=0$ case? Next consider the case $n=1$, especially when the filler $\Delta^1\to X$ is an identity edge. The cases $n=0,1$ are all that are needed for, and are equivalent to, your result.