I asked this question already on math.stackexchange, but maybe it is also useful to ask this here, since it was not answered there.

Suppose we have three directed sequences of $C^*$-algebras, say $(A_n,\varphi_n)$,$(B_n,\psi_n)$ and $(C_n,\theta_n)$ and $*$-homomorphisms $\alpha_n:A_n\rightarrow C_n$ and $\beta_n:B_n\rightarrow C_n$, then we can take the pullback $A_n\times_{C_n}B_n$ for all $n\in\mathbb{N}$ and can also take the direct limit, thus $\lim_{n\rightarrow\infty}{A_n\times_{C_n}B_n}$. My question is: Does the following hold: $$\lim_{\rightarrow}{A_n\times_{C_n}B_n}=\lim_{\rightarrow}{A_n}\times_{\lim_{\rightarrow}{C_n}}\lim_{\rightarrow}{B_n}$$ or in other words: do direct limits preserve pullbacks? From my point of view this does not hold in general but I can not find a good argument. Someone an idea? Thank you very much.