The Clowder Project

\[ \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (x_{\alpha }\webright )_{\alpha \in I}\in \prod _{\alpha \in I}X_{\alpha }\ \middle |\ \begin{aligned} & \text{for each $\alpha ,\beta \in I$, if $\alpha \preceq \beta $,}\\ & \text{then we have $x_{\alpha }=f_{\alpha \beta }\webleft (x_{\beta }\webright )$} \end{aligned} \webright\} . \]