Preservation of $(2, 1)$-colimits under the nerve

Question

Let $\mathcal{Cat}$ denote the $(2, 1)$-category of categories, functors, and natural isomorphisms, understood as an $(\infty, 1)$-category. Say we have a small $1$-category $\mathcal{J}$ and a diagram $F : \mathcal{J} \to \mathcal{Cat}$ which commutes strictly. The nerve construction gives an $(\infty, 1)$-functor $N : \mathcal{Cat} \to (\infty, 1)\mathcal{Cat}$. There is then a colimit comparison morphism $\mathrm{colim} (N \circ F) \to N(\mathrm{colim} F)$ in $(\infty, 1)\mathcal{Cat}$. Is this map an isomorphism? If not in general, is it an isomorphism when $F$ is a span of categories?

Answer 1

For a counterexample, consider the span of categories $\def\B{{\sf B}}\def\Z{{\bf Z}}1←\B\Z=S^1→1$, where 1 denotes the terminal groupoid and $\B$ denotes the delooping functor.

The colimit of this span in the (2,1)-category Cat is the terminal groupoid 1.

The colimit of this span in the (∞,1)-category (∞,1)Cat is the (∞,1)-groupoid $S^2$.