(Added the "dual" tick in the LHS of the last line in Mrc Plm's good answer.) For someone who hadn't thought in these terms before, it is probably worth noting, further to Mrc Plm's, that the duals to projective limits $B={\rm projlim}_j B_j$ are not determined purely categorically, but must be "computed" a little, by proving that any TVS hom of such a projective limit to a _normed_ space (such as scalars) must factor through a limitand.

In contrast, that the dual of a colimit is the corresponding limit of duals _is_ formal.

Also, as in Mrc Plm's answer, while indeed $C^\infty(X)=\lim_k C^k(X)$ is complete-metrizable,  the metrizability of its dual is less obvious to me. The dual of $C^o(X)$ is compactly-supported measures, etc., but I myself hadn't had occasion to think of a metric on that colimit, nor whether it'd be _complete_, which in my experience is the salient issue.