connected components of homotopy pullbacks of nerves of categories I am not sure if the question makes sense.
Given three categories $ M_1$, $M_2$ and $ M_3$ and a zig-zag $$M_1\xrightarrow{f} M_3\xleftarrow{g} M_2$$
Let $ 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 $m_1\in M_1$,$m_2\in M_2$, $m_3\in M_3$ 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 $ N$ is the homotopy pullback of the nerve of the zig-zag $ M_1\xrightarrow{f}  M_3\xleftarrow{g} M_2$.
My question is, is it true that (no condition is required) the set of connected components of $ 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 $ D_1$, $ D_2$, $ D_3$ and $ 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 $ C-Mod$ denote the category of DG $ C-$modules. This is a $C(k)-$model category with the obvious $C(k)-$enrichment. Consider the category $ M( C, D_i)$ of DG $ C\otimes D_i^{\mathrm{op}}-$modules $X$ such that for each $c\in C$, $X(c,-)$ is quasi-isomorphic to a representable $ D^{\mathrm{op}}-$module, with morphisms quasi isomorphisms of DG modules. Let $ D_1\xrightarrow{u}  D_3\xleftarrow{v} D_2$ be a zig-zag of DG categories such that $u:D_1\rightarrow  D_3$ is a fibration. Then we have the induced diagram of categories : $$ M( C, D_1)\rightarrow M( C,D_3)\leftarrow  M( C, D_2) $$
Let $ N$ be as above. Then Toen says it is easy to see that the set of connected components of the nerve of $ N$ is isomorphic to that of the homotopy pullback of the zig-zag.
 A: An answer to Toen's claim is as follows:
First, we replace the categories $M(C,D_i)$ with the more rigid categories (because we are really using morphisms in the dg module categories rather than in the homotopy categories), namely, we take the full subcategories of $M(C,D_i)$ consisting of cofibrant bimodules. Note that these full subcategories are equivalent to the ambient categories. We keep the same notation for these full subcategories. Notice also that the left Quillen functors $p_{!}$ and $q_{!}$ preserve these subcategories.
Recall we are considering the co-span of dg categories
$$
D_1\xrightarrow{u}D_3\xleftarrow{v}D_2.
$$
Claim: for $i=1,2$, the zig-zags
$$
u_{!}X_i\rightarrow Y_i \leftarrow v_{!}Z_i
$$
are in the same connected component if and only if $[X_0]=[X_1]$ in $\pi_0(N(M(C,D_1)))$, $[Y_0]=[Y_1]$ in $\pi_0(N(M( C,D_3)))$ and $[Z_0]=[Z_1]$ in $\pi_0(N(M(C,D_2)))$.
Indeed we need the fact that the adjunction $(u_!,u^*): C_{dg}(C\otimes D_1)\rightarrow C_{dg}(C\otimes D_2)$ is an enriched adjunction.
Since all objects are cofibrant, we have a commutative diagram up to homotopy
\begin{aligned}
&u_!X_1\rightarrow Y_1\leftarrow v_!Z_1\\
&\downarrow    \ \ \ \ \ \ \ \ \ \downarrow\ \ \ \ \ \ \ \downarrow\\
&u_!X_2\rightarrow Y_2\leftarrow v_! Z_2
\end{aligned}
We consider the left square which is commutative up to homotopy with homotopy of the form $d(h)$ for some $h:u_!X_1\rightarrow Y_2$ of degree -1.
Then by adjunction, we have a morphism $h':X_1\rightarrow u^*Y_2$ of degree -1 which corresponds to $h$. Let $I X_1$ be the
mapping cone of $\mathrm{Id}_{X_1}$. This dg module lies in $M(C,D_1)$. And the morphism $h':X_1\rightarrow u^* Y_2$ factors through $IX_1$. So we have a genuine commutative diagram in the category of dg modules
$$
\ \ \ \ X_1\longrightarrow\ \ \ \ \ \ \ \ \ \ u^*Y_1\\
\downarrow\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \downarrow\\
X_2\oplus IX_1\rightarrow u^*Y_2\\
\uparrow\ \ \ \ \ \ \ \ \uparrow\\
X_2\rightarrow \ \ \ \ u^*Y_2
$$
Similarly for the right hand square. By adjunction we yield a zig-zag connecting the two objects and thus they are in the same connected component. This proves the Claim.
Now we take a Kan replacement of $N(M(C,D_3))$:
$$N(M(C,D_3))\xrightarrow{r} K_3$$
where $K_3$ is a Kan complex and $r$ is a trivial cofibration in the Kan-Quillen model structure.
Then we take trivial cofibration- fibration factorizations
$$N(M(C,D_1))\xrightarrow{p} K_1\xrightarrow{f} K_3\\
N(M(C,D_2))\xrightarrow{q} K_2\xrightarrow{g} K_3 
$$
of the functors
$$N(M(C,D_1))\xrightarrow{u} N(M(C,D_3))\xrightarrow{r} K_3\\
N(M(C,D_2))\xrightarrow{v} N(M(C,D_3))\xrightarrow{r} K_3
$$.
The pullback $K$ of
$$K_1\xrightarrow{f} K_3\xleftarrow{g}K_2$$
is the required homotopy pullback.
We construct a map sending the connected component of a 5-tuple
$(d_1,d_2,d_3,u(d_1)\rightarrow d_3,v(d_2)\rightarrow d_3)$,
where $d_i\in N(M(C,D_i))$ and $u(d_1)\rightarrow d_3\leftarrow v(d_2)$ are maps in $N(M(C,D_3))$,
to $[(d_3',d_3'')]\in \pi_0(K)$ where we have morphisms $p(d_1)\rightarrow d_3'$ whose image under $f$ is the map $fp(d_1)=ru(d_1)\rightarrow r(d_3)$ induced by $r$ and $q(d_2)\rightarrow d_3''$ whose image under $g$ is the map $gq(d_2)=rv(d_2)\rightarrow r(d_3)$ induced by $r$ (note that $f$ and $g$ are Kan fibrations). It is clear that this is a well-defined map. We leave it as an exercise to check that this is a surjective map.
To show that this is an injective map, we use the claim.
