Given a category $\mathcal{C}$ enriched in spaces, we can take the nerve (a simplicial space) and then geometric realization to get a space $B\mathcal{C}$. If we view spaces as $\infty$-groupoids, then this process should be thought of as a ($\infty$-)groupoidification.

We can also consider the homotopy category $h\mathcal{C}$, which has the same objects as $\mathcal{C}$ but where the morphisms from x to y are given by $\pi_0 \mathcal{C}(x,y)$. This is an ordinary category and we can take the nerve and geometrically realize it to get the classifying space $Bh\mathcal{C}$.

In general the spaces $B\mathcal{C}$ and $Bh\mathcal{C}$ will be very different, but they might agree on some low dimensional homotopy groups.

Fix an object $x \in \mathcal{C}$. Is it true that $\pi_1(B\mathcal{C}, x)$ is isomorphic to $\pi_1( Bh\mathcal{C}, x)$? If not, what is a good counter example? Are there conditions under which these will be isomorphic? For example I am interested in the case where $\mathcal{C}$ is symmetric monoidal and $x$ is the unit object.

Note that we can view a set as a discrete topological space and so $h\mathcal{C}$ is also a (discrete) topological category. There is a functor $\mathcal{C} \to h\mathcal{C}$, and so there is a natural comparison map $\pi_1(B\mathcal{C}, x)\to\pi_1( Bh\mathcal{C}, x)$.