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.


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