Dimitri Ara has just brought this question to my attention. Perhaps you will find the following useful. 

Let us say that an object $z$ of a $2$-category $\mathcal{A}$ has a terminal object if, for every object $x$ of $\mathcal{A}$, the category $Hom_{\mathcal{A}}(x,z)$ has a terminal object. (In particular, this category is non-empty.) 

The terminology comes from the fact that a category has a terminal object in this sense, viewed as an object of the $2$-category of categories, if and only if it has a terminal object in the usual sense. (This terminology was suggested to me by Bénabou because of that. See also Michael Barr's relevant answer to my question https://mathoverflow.net/questions/108397/is-there-a-standard-name-for-a-2-category-which-has-an-object-z-such-that-for-e) 

There are three dual definitions obtained by replacing $Hom_{\mathcal{A}}(x,z)$ by $Hom_{\mathcal{A}}(z,x)$ or "terminal" by "initial". The result below could of course be dualized accordingly. 

Denote by $e$ the trivial $2$-category. If a small $2$-category $\mathcal{A}$ has an object which has a terminal object, then the geometric realization of $\mathcal{A} \to e$ is a homotopy equivalence. This is Lemme 2.27 of http://www.math.jussieu.fr/~chiche/Maths/TheoremA.pdf. I still have to upload the final version on Arxiv. This is to be published in TAC.

Not sure whether it answers the question you were asking but hope this helps nevertheless.