  We consider a family $\pi^i_j\colon B_i\rightarrow B_{ij}=B_{ji}$, $i,j\in
\{1,2,3\}$, $i\neq j$, of $C^*$-epimorphisms assuming that it satisfies the
cocycle condition. Then we show how to compute the $K$-groups of the
multi-pullback $C^*$-algebra of such a family, and examplify it in the case of
the triple-Toeplitz deformation of $\mathbb{C}P^2$.
