In the nlab entry on uniform spaces they speak about an "inherited uniform structure on function spaces". Namely, if $X$ is a set and $(Y,\mathfrak{U})$ is a uniform space, then $Y^X$ can be equipped with the uniform structure generated by: $$ \bigg\{\big\{ (f,g) \,|\, (f(x),g(x))\in U \;\forall x\in X \big\}, U\in \mathfrak{U}\bigg\}\;. $$

Is then this the same as the uniform structure on the $X$-fold Cartesian product? If not, does it hold for $X$ "small" enough, e.g. finite?