This paper extends a model theory for a family of not necessarily commuting operators developed by A. E. Frazho [J. Funct. Anal. 48, 1-11 (1982; Zbl 0487.47011)] and J. W. Bunce [J. Funct. Anal. 57, 21-30 (1984; Zbl 0558.47004)]. The author poves that for any doubly indexed family \(\{A_{\alpha,i}:\alpha\in J\), \(i\in \Lambda \}\) (card\(\Lambda\leq \aleph_ 0)\) of operators on a Hilbert space \({\mathcal H}\) such that \(\sum_{i\in \Lambda}A^*_{\alpha,i}A_{\alpha,i}\leq I\) for each \(\alpha\in J\), there exists a family \(\{V_{\alpha,i}:\alpha\in J\), \(i\in \Lambda \}\) of coisometries on a Hilbert space \({\mathcal K}\) (\(\supseteq {\mathcal H})\) such that \(\sum_{i\in \Lambda}V^*_{\alpha,i}V_{\alpha,i}\leq I,\) \(V_{\alpha,i}{\mathcal H}\subseteq {\mathcal H}\) and \(V_{\alpha,i}| {\mathcal H}=A_{\alpha,i}\) for each \(\alpha\in J\), \(i\in \Lambda\). The coisometry \(V_{\alpha,i}\) can be required to be a pure cosiometry for \(\alpha\in J\) and \(i\in \Lambda\) for which \(A^ n_{\alpha,i}\to 0\) (strongly) as \(n\to \infty\). Then he gives conditions for a family of operators to be simultaneously similar to a family \(\{T_ i\}_{i\in \Lambda}\) of contractions with \(\sum_{i\in \Lambda}T^*_ iT_ i\leq I\), in particular, he proves versions of the B. Sz.-Nagy theorem [Acta Sci. Math. 11, 152-157 (1947; Zbl 0029.30501)] and the G. C. Rota model theorem [Commun. Pure Appl. Math. 13, 469-472 (1960; Zbl 0097.316)] for a family of operators.