Let $\mathbf{C}$ be a small category and $\mathbf{C}'$ its [hammock localization](http://ncatlab.org/nlab/show/simplicial+localization#hammock_localization) in the sense of Dwyer and Kan. I am looking for a proof (or counterexample) of the following assertion:



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> If there is an initial object (up to homotopy) in $\mathbf{C}'$, then $\mathbf{C}$ has a contractible classifying space. 
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By initial up to homotopy I mean an object $c$ so that $\mathbf{C'}(c,d)$ is contractible for all other objects $d$.



Zigzags and contractibility of categories
The answer to your new question is positive. Dwyer and Kan prove in *Calculating simplicial localizations* that the hammock localization ${\bf C}’=L^H\bf C$ is weakly equivalent to the standard simplicial localization $L\bf C$. In *Simplicial localizations of categories*, they had previously showed that $L\bf C$ and $\bf C$ have weakly equivalent nerves. The nerve functor preserves weak equivalences of simplicial categories with the same objects (see *Simplicial localizations of categories* 1.4 (vii)), so it is enough to prove that $NL\bf C$ is contractible. Since you’re taking hammock/simplicial localization w.r.t. all arrows in $\bf C$, $L\bf C$ is a simplicial groupoid by its very definition. The existence of a homotopy initial object, which is transferred to $L\bf C$ by the weak equivalence above, implies that *all* simplicial sets of morphisms in $L\bf C$ are contractible. Therefore, $L\bf C$ has the same nerve as the trivial groupoid on the same set of objects as $\bf C$, which is of course contractible.


