I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the presence of initial objects in all the fibers $d\setminus F$ and use the 2-categorical version of Quillen's Theorem A due to Bullejos and Cegarra (see their pdf here). It'd be nice if these fibers had initial elements, so
What is an initial object $i$ in a 2-category $C$, and where can I find a reference in the literature?
I imagine that instead of having a unique morphism $1 \to c$ for every object $c$ of $C$ like one does for a 1-categorical initial object, we'd now want the category of all morphisms from $1$ to $c$ in $C$ to be contractible. So if $C$ is poset-enriched it would suffice for the poset $C(i,c)$ to have a minimal element for all objects $c$. Is this accurate, and if so, what can I cite as a reference?