Proposition 5.6.4. Let $X, Y$ be uniform spaces, $X', Y'$ be their associated Hausdorff uniform spaces, and $\wh X, \wh Y$ be their Hausdorff completions, then there exists a unique $F \in UC(X'; Y')$ and $\ol F \in UC(\wh X; \wh Y)$ such that the following diagram commutes:
\[\xymatrix{
\wh X \ar@{->}[r]^{\overline{F}} & \wh Y \\
X' \ar@{->}[u] \ar@{->}[r]^{F} & Y' \ar@{->}[u] \\
X \ar@{->}[r]_{f} \ar@{->}[u] & Y \ar@{->}[u]
}\]
Proof. By (U) of Definition 5.6.1 and Definition 5.6.3.$\square$