Definition 10.88.1. Let $(M_ i, f_{ij})$ be a directed system of $R$-modules. We say that $(M_ i, f_{ij})$ is a *Mittag-Leffler directed system of modules* if each $M_ i$ is an $R$-module of finite presentation and if for every $R$-module $N$, the inverse system

\[ (\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, N), \mathop{\mathrm{Hom}}\nolimits _ R(f_{ij}, N)) \]

is Mittag-Leffler.

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