Corollary

Let $(\mathrm{F}_\alpha, g_{\beta\alpha})$ and $(\mathrm{F}'_\alpha, g'_{\beta\alpha})$ be two direct systems of sets relative to I, and for each $\alpha \in \mathrm{I}$ let $u_\alpha : \mathrm{E}_\alpha \to \mathrm{F}_\alpha$, $u'_\alpha : \mathrm{E}'_\alpha \to \mathrm{F}'_\alpha$ be mappings such that $(u_\alpha)$ and $(u'_\alpha)$ are two direct systems of mappings. Then $(u_\alpha \times u'_\alpha)$ is a direct system of mappings, and (up to canonical bijections) we have

(35)

We leave the verification to the reader.