Let $M_X = (X, \mathcal{F}_X)$ and $M_Y = (Y, \mathcal{F}_Y)$ each be a D1108: Measurable space.
A D18: Map $f : X \to Y$ is measurable with respect to $M_X$ and $M_Y$ if and only if \begin{equation} \sigma_{\text{pullback}} \langle f \rangle \subseteq \mathcal{F}_X \end{equation}

Let $M_X = (X, \mathcal{F}_X)$ and $M_Y = (Y, \mathcal{F}_Y)$ each be a D1108: Measurable space.
A D18: Map $f : X \to Y$ is a measurable map from $M_X$ to $M_Y$ if and only if \begin{equation} \forall \, E \in \mathcal{F}_Y : f^{-1}(E) \in \mathcal{F}_X \end{equation}