A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal precategory satisfying the Segal condition.

Namely, we want the natural maps $$\phi_k:X_k \to X_1 \times_{X_0} \cdots \times_{X_0} X_1$$ to be weak equivalences $\forall k\geq2$.

There are several reasons for requiring the codomain to be a homotopy limit, and in the analogous case of Segal spaces we see immediately that this holds.

My question is: why does the discreteness of $X_0$ imply the codomain of $\phi_k$ is a homotopy limit? I guess it may be useful to know that $X_0$ is automatically fibrant.

Thanks in advance for any hint!