Suppose that $\mathscr{C}$ is a $2$-category (or more generally a bicategory) which is not a $\left(2,1\right)$-category. Is there any relation between limits and colimits in $\mathscr{C}$ (in the **weak** $2$-categorical sense), and limits and colimits in the associated $\left(\infty,1\right)$-category? By the associated $\left(\infty,1\right)$-category, I mean any fibrant replacement of its homotopy coherent nerve (viewing $\mathscr{C}$ as a simplicial category) in the Joyal model structure, or in the bicategory case, the Duskin nerve of $\mathscr{C}$.

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don'twant to do. My construction isn't "incorrect", I'm just after a different thing. I've constructed a bicategory where the mapping categories are supposed to model the homotopy type (via their classifying space) of the mapping spaces in the $(\infty,1)$-category I'm really after. $\endgroup$ – David Carchedi Aug 21 '14 at 10:56somerelation between limits and colimits in the bicategory and in the $(\infty,1)$-category. They certainly won't bethe same- that would be way too strong. I'm just wondering if there's any relation at all.shrug? $\endgroup$ – David Carchedi Aug 21 '14 at 12:47