The Cone.The collection
\[ \webleft\{ \text{pr}_{\gamma }\colon \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )\to X_{\gamma }\webright\} _{\gamma \in I} \]
of maps of sets defined as the restriction of the maps
\[ \webleft\{ \text{pr}_{\gamma }\colon \prod _{\alpha \in I}X_{\alpha }\to X_{\gamma }\webright\} _{\gamma \in I} \]
of Item 2 of Construction 2.1.2.1.2 to $\displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$ and hence given by
\[ \text{pr}_{\gamma }\webleft (\webleft (x_{\alpha }\webright )_{\alpha \in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{\gamma } \]
for each $\gamma \in I$ and each $\webleft (x_{\alpha }\webright )_{\alpha \in I}\in \displaystyle \operatorname*{{\displaystyle \underset {\longleftarrow }{\operatorname*{\text{lim}}}}}_{\alpha \in I}\webleft (X_{\alpha }\webright )$.