Let $M_j = (Y_j, \mathcal{F}_j)$ be a D1108: Measurable space for each $j \in J$ such that

(i)	$f_j : X \to Y_j$ is a D18: Map from $X$ to $Y_j$ for each $j \in J$

The pullback sigma-algebra on $X$ with respect to $f = \{ f_j \}_{j \in J}$ and $M = \{ M_j \}_{j \in J}$ is the D11: Set $$\begin{equation} \sigma \left\langle \bigcup_{j \in J} \left\{ f^{-1}_j(E) : E \in \mathcal{F}_j \right\} \right\rangle \end{equation}$$