# Kerodon

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Example 11.10.3.7. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be unitary lax functor, let $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the Grothendieck construction on $\mathscr {F}$ (Construction 11.5.0.79), and let $U: \int ^{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ be the forgetful functor. Suppose we are given a morphism $(f,u): (C,X) \rightarrow (D,Y)$ in the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$, so that $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $u: X \rightarrow \mathscr {F}(f)(Y)$ is a morphism in the category $\mathscr {F}(C)$. For any other object $X' \in \mathscr {F}(C)$, postcomposition with $(f,u)$ determines a function

$\theta : \operatorname{Hom}_{ (\int ^{\operatorname{\mathcal{C}}} \mathscr {F})_{C} }( X', X ) \rightarrow \operatorname{Hom}_{ \int ^{\operatorname{\mathcal{C}}} \mathscr {F} }( (C,X'), (D,Y) ) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) } \{ f\} .$

Using the assumption that $\mathscr {F}$ is unitary (together with Remark 5.6.1.12), we can identify $\theta$ with the map $\operatorname{Hom}_{\mathscr {F}(C)}( X', X) \rightarrow \operatorname{Hom}_{\mathscr {F}(C) }( X', \mathscr {F}(f)(Y) )$ given by postcomposition with $u$. It follows that $(f,u)$ is a locally $U$-cartesian morphism of the category $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ if and only if $u$ is an isomorphism in the category $\mathscr {F}(C)$.