Proposition 5

Let $X$ be a set, let $(Z_i)_{i \in I}$ be a family of uniform spaces, let $(J_\lambda)_{\lambda \in L}$ be a partition of $I$ and let $(Y_\lambda)_{\lambda \in L}$ be a family of sets indexed by $L$. For each $\lambda \in L$, let $h_\lambda$ be a mapping of $X$ into $Y_\lambda$; for each $\lambda \in L$ and each $i \in J_\lambda$, let $g_{i\lambda}$ be a mapping of $Y_\lambda$ into $Z_i$; and let $f_i = g_{i\lambda} \circ h_\lambda$. Let each $Y_\lambda$ carry the coarsest uniform structure for which the mappings $g_{i\lambda}$ ($i \in J_\lambda$) are uniformly continuous. Then the coarsest uniform structure on $X$ for which the $f_i$ are uniformly continuous is the same as the coarsest uniform structure on $X$ for which the $h_\lambda$ are uniformly continuous.