(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.
Edited: Also, as in Mrc Plm's answer, indeed $C^\infty(X)=\lim_k C^k(X)$ is complete-metrizable. Metrizability of the dual is subtler. One should know that the dual of $C^o(X)$ is compactly-supported measures, etc. (Also, whether or not a topology is metrizable, completeness is often the salient issue.)