Proposition B.8

For every index type $I$ in universe $u$ and every family $(Y_i)_{i\in I}$ of spaces in universe $u$ compactly generated with respect to compact Hausdorff spaces in universe $u$, the cone with vertex $k\bigl(\prod_{i\in I}Y_i\bigr)$ and coordinate projections is a limit cone in the category of such spaces. Here the product initially has its usual product topology, and its refinement $k\bigl(\prod_iY_i\bigr)$ declares a subset open exactly when every continuous map from a compact Hausdorff space in universe $u$ into the original product has open inverse image of that subset. Explicitly, for every compactly generated space $A$ in this category and every family of continuous maps $f_i:A\to Y_i$, there exists exactly one continuous map $f:A\to k\bigl(\prod_iY_i\bigr)$ whose $i$-th coordinate is $f_i$ for every $i$. This includes empty $I$, for which the vertex is a singleton and is terminal in the category.