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Let $P_{i,j,k}=\{1_{i,j,k},\ldots,n_{i,j,k},\ldots,\gamma(i,j,k)_{i,j,k}\}$ (so that we have a distinct set of size $\gamma(i,j,k)$, one for each choice of $i,j,k$). Let $Q=\{p_k\mid k\in K\}$ (so just $K$, up to relabeling). Let $R=\{\ast\}$. Let $$X=Q\coprod R\coprod_{i\in I,j\in J,k\in K}P_{i,j,k}.$$ Define $$\Omega_j=\coprod_{i\in I,k\in K}P_{i,j,k}\subset X,$$ and $$f_{i}(n_{i_0,j_0,k_0})=\begin{cases}p_{k_0}\text{ if }i=i_0\\\ \ast\text{ otherwise}\end{cases}$$ $$f_i(p_k)=\ast\hskip0.3in f_i(\ast)=\ast$$ Thus $$f_i^{-1}(p_k)=\coprod_{j\in J}P_{i,j,k}\subset X,$$ and thus $f_i^{-1}(p_k)\cap \Omega_j=P_{i,j,k}$, so $|f_i^{-1}(p_k)\cap\Omega_j|=\gamma(i,j,k)$.