Kerodon

Example 9.2.4.3 (Flat Modules as Flat Functors). Let $R$ be an associative ring and let $\operatorname{\mathcal{C}}$ be the category of free left $R$-modules of finite rank. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is flat if and only if it has the form we see that a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is flat if and only if it has the form $\operatorname{Hom}_{R}( \bullet , M)$, where $M$ is a flat left $R$-module. This is a reformulation of Lazard's characterization of flat modules (Variant 9.2.3.9), by virtue of Remark 9.2.4.6 below.