product of locally compact spaces preserves colimits
1. Proposition
Let \((X, \mathcal{T})\) be a locally compact topological space. Then the functor
\begin{align*} X \times - : \mathrm{Top} \rightarrow& \mathrm{Top} \\ Y \mapsto& X \times Y \\ (f: Y \rightarrow Z) \mapsto& ((\mathrm{id}_{X}, f): X \times Y \times X \times Z) \end{align*}