Homotopy category of a simplicial category

Question

In many places (for example here) I've seen the following definition:

For a simplicial category $\mathcal{C}$, it's homotopy category is defined to be the category $Ho(\mathcal{C})$ with the same objects as $\mathcal{C}$ and with morphisms $$Ho(\mathcal{C})(x,y)=\pi_0\mathcal{C}(x,y).$$ The problem I have with this definition is that homotopy groups of simplicial sets (in particular $\pi_0$) are well-defined only for Kan complexes.

So why does this definition work? Is $\mathcal{C}(x,y)$ necessarly a Kan complex for every $x,y\in \mathcal{C}$? Do we take fibrant replacement without loss of generallity?

Answer 1

$\pi_0$ is well-defined for any simplicial set $S$: it's the coequalizer of the two face maps $S_1 \to S_0$. This is the left adjoint to the inclusion of sets into simplicial sets.