I was going back through some class notes discussing the direct limit topology (final topology) and we showed that the direct limit of topological groups is a topological group. To do this, we showed that $\varphi: (x,y) \mapsto xy^{-1}$ is continuous. Let $\displaystyle X = \lim_{\rightarrow} X_{s}$. To show that $\varphi$ is continuous, we were working in (what I wrote down as) $(X \times X)_{s}$ and then took the direct limit. This is where I get confused.
How does one put a direct systems on a product of direct limits?