My question refers to a statement in Laures' and Szymik's "Grundkurs Topologie" (page 233). Sorry, there exist only a German version. Here the relevant excerpt:
My question is why a category having a initial (or a final) object has a contractible classifying space?
Previously the author just shows that equivalent categories induce homotopy equivalent classifying spaces via
$$BC\times \Delta^1 \to BD$$.
Does this observation provide an answer of my question? Why?
Here the construction of classifying space with which the author works:, namely as a nerve of the functor $B:Cat \to \Delta^{op}-Sets, C \mapsto Mor(-,C)$:
