Proposition 5.2.7. Let $X$, $\seqi{Y}$ be uniform spaces, $\seqi{f}$ where $f_{i} \in UC(X; Y_{i})$ for each $i \in I$, then
There exists a unique $f \in UC(X; \prod_{i \in I}Y_{i})$ such that the following diagram commutes
\[\xymatrix{ & \prod_{i \in I} Y_i \ar@{->}[rd]^{\pi_i} & \\ X \ar@{->}[rr]_{f_i} \ar@{->}[ru]^{f} & & Y_i }\]for all $i \in I$.
If $X$ is T0 and equipped with the initial uniformity induced by $\seqi{f}$, then $f$ is an isomorphism onto its image.
Proof. (1): By (U) of Definition 5.2.5.
(2): By (2) of Proposition 4.7.3, $f$ is injective.
Let $U$ be an entourage in $X$. Assume without loss of generality that there exists $J \subset I$ finite and $\bracs{U_j}_{j \in J}$ such that $U = \bigcap_{j \in J}(f_{j} \times f_{j})^{-1}(U_{j})$. In which case, $\bigcap_{j \in J}(\pi_{j} \times \pi_{j})^{-1}(U_{j})$ is open in $\prod_{i \in I}Y_{i}$ and