Let $sFre\_{\mathbb{R}}$ *(resp. $Fre\_{\mathbb{R}}$)* denote the category of *(resp. separable)* Fr\'{e}chet spaces over $\mathbb{R}$ as objects, and bounded linear operators as morphisms. 


Is this a topological concrete category over $\mathbb{R}-Vect$; in the sense of [this post](https://ncatlab.org/nlab/show/topological+concrete+category). In particular, is it [cototal](https://ncatlab.org/nlab/show/total%20category)?



Category of Frechet Spaces is Topological?
Posted as an answer at the OP‘s request: Since the category of Fréchet spaces doesn‘t admit products and sums in general, the answer would appear to be negative. This is analogous to the fact that the category of metric spaces is apparently not topological whereas that of uniform spaces is (for similar reasons).


