Let $M_j = (X_j, \mathcal{F}_j)$ be a D1108: Measurable space for each $j \in J$.
Let $f_j : X_j \to Y$ be a D18: Map for each $j \in J$.
The pushforward sigma-algebra on $Y$ with respect to $f = \{ f_j \}_{j \in J}$ and $M = \{ M_j \}_{j \in J}$ is the D11: Set \begin{equation} \bigcap_{j \in J} \{ E \subseteq Y \mid f^{-1}_j(E) \in \mathcal{F}_j \} \end{equation}