Let $D \colon \mathbf{J} \to \mathbf{Cat}$ be a filtered diagram of categories and functors. It has a colimit $\mathbf{C} = \mathrm{colim}\;D$. If you replace the diagram by a naturally isomorphic one $D' \colon \mathbf{J} \to \mathbf{Cat}$, then the colimit 
$\mathbf{C'} = \mathrm{colim}\;D'$ is isomorphic to $\mathbf{C}$. What happens if you replace the diagram by a naturally *equivalent* one $D'' \colon \mathbf{J} \to \mathbf{Cat}$, is the colimit $\mathbf{C''} = \mathrm{colim}\;D''$ still equivalent to $\mathbf{C}$?

(I'm interested in $\mathbf{Cat}$, or in fact the category $\mathbf{MonCat}$ of monoidal categories, but suspect that there is a general statement in weak 2-categories. Any references are appreciated.)