Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}$, let $f: U \to V$ be a morphism in $\mathcal{C}$, and let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(U))$. A

*pushforward*of $x$ along $f$ is a morphism $x \to y$ of $\mathcal{F}$ lying over $f$. A pushforward is unique up to unique isomorphism (see the discussion following Categories, Definition 4.33.1). We sometimes write $x \to f_*x$ for “the” pushforward of $x$ along $f$.

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