In our lecture we defined a category $\mathcal C$ to have arbitrary products, iff every diagram $A:\mathcal J \to \mathcal C$ with $\text{Morph}(\mathcal J)=\{\text{id}_j\}_{j \in Ob(\mathcal J)}$ has a limit. We showed that the category Man (with manifolds modelled on locally convex spaces) possesses finite products (i.e. $Ob(\mathcal J)$ finite), but argued that it does not possess infinite ones. As far as i can see, in defiance of this in the following we always use the product over $Ob(\mathcal J) = \mathbb N$.
My question is now: Is the infinite (countable) cartesian product of manifolds still a manifold? If not, what is the problem? Does the category of locally convex spaces Lcs possess arbitrary products?