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Of course you are right. Thanks. The path is one in the one skeleton of $B(\mathcal{C})$ in which some identifications took place like $[(x\rightarrow y),(0,1)]=[(z\rightarrow y),(1,0)]$ in your example. One can figure out that iterations of this case are the only thing which can happen, so we get a zig-zag as desired.
@ZhenLin The nLab (ncatlab.org/nlab/show/connected+space#locally_connected_spaces) says locally connected spaces form a coreflective subcategory. Does that help to conclude your statement if $ob\mathcal{D}$ and $mor\mathcal{D}$ are assumed to be locally connected?
I would accept to restrict to categories $\mathcal{D}$ which objects and morphisms are locally connected or other convenient point-set topological restrictions, as long as all the constructions (taking pullbacks for the nerve, etc.) find place in the category of all topological spaces, since I don't want to get another space $B\mathcal{D}$ by changing the topological category.
