 The \(j\)-th operation of \(\Pi A_i\), of arity \(n_j\), takes an \(n_j\)-tuple \(t\) of elements of \(\Pi X_i\) and produces the \(I\)-tuple \(\langle f_{j}^i(t_{1}^i , \ldots t_{n_{ j} }^i)\rangle_{i\in I}\) where \(t_k^i\) is the \(i\)-th component of the \(k\)-th component of \(t\) for \(k\) from 1 to \(n_j\).