# connected components of homotopy pullbacks of nerves of categories

I am not sure if the question makes sense.

Given three categories $$\mathcal M_1$$, $$\mathcal M_2$$ and $$\mathcal M_3$$ and a zig-zag $$\mathcal M_1\xrightarrow{f} \mathcal M_3\xleftarrow{g} \mathcal M_2$$ Let $$\mathcal N$$ be the category of five-tuples $$(m_1,m_2,m_3;a:f(m_1)\rightarrow m_3,b:g(m_2)\rightarrow m_3)$$ where $$m_1\in \mathcal M_1$$,$$m_2\in \mathcal M_2$$, $$m_3\in \mathcal M_3$$ and the morphisms are the obvious one (making the obvious diagram commute).

Under a so-called 'quasi-fibrancy' condition (Finite homotopy limits of nerves of categories, page 3, remark after the corollary), the nerve of the category $$\mathcal N$$ is the homotopy pullback of the nerve of the zig-zag $$\mathcal M_1\xrightarrow{f} \mathcal M_3\xleftarrow{g} \mathcal M_2$$.

My question is, is it true that (no condition is required) the set of connected components of $$\mathcal N$$ is always isomorphic to that of the homotopy pullback of the zig-zag? Or is there a weaker condition to assure that this is true?

Edit: Probably I should say one example. Let $$\mathcal D_1$$, $$\mathcal D_2$$, $$\mathcal D_3$$ and $$\mathcal C$$ be DG categories. Tabuada has constructed a cofibrantly generated model structure on the category of DG categories (over a fixed commutative ring $$k$$) where the weak equivalences are the quasi equivalences (cf. The homotopy theory of dg-categories and derived morita theory,Definition 2.1). Let $$\mathcal C-Mod$$ denote the category of DG $$\mathcal C-$$modules. This is a $$C(k)-$$model category with the obvious $$C(k)-$$enrichment. Consider the category $$\mathcal M(\mathcal C,\mathcal D_i)$$ of DG $$\mathcal C\otimes\mathcal D_i^{\mathrm{op}}-$$modules $$X$$ such that for each $$c\in C$$, $$X(c,-)$$ is quasi-isomorphic to a representable $$\mathcal D^{\mathrm{op}}-$$module, with morphisms quasi isomorphisms of DG modules. Let $$\mathcal D_1\xrightarrow{p} \mathcal D_3\xleftarrow{q} \mathcal D_2$$ be a zig-zag of DG categories such that $$p:\mathcal D_1\rightarrow \mathcal D_3$$ is a fibration. Then we have the induced diagram of categories : $$\mathcal M(\mathcal C,\mathcal D_1)\rightarrow\mathcal M(\mathcal C,\mathcal D_3)\leftarrow \mathcal M(\mathcal C,\mathcal D_2)$$ Let $$\mathcal N$$ be as above. Then Toen says it is easy to see that the set of connected components of the nerve of $$\mathcal N$$ is isomorphic to that of the homotopy pullback of the zig-zag.