For holomorphic line bundle we define its first Chern class by exponential sequence $$0\to \mathbb Z \to \mathcal O \to \mathcal O^* \to 0 $$ and we can similarly define Chern class for smooth line bundle by the short exact sequence $$0\to \mathbb Z \to \mathcal C^\infty \to (\mathcal C^\infty)^*\to 0$$
Then there is a natural morphism from the first short exact sequence to the second one, so there is a natural map $H^2(\mathbb Z)\to H^2(\mathbb Z)$. Is this map isomorphic? Similarly, is the map $H^1(\mathcal O^*)\to H^1((\mathcal C^\infty)^*)$ just the natural map of on equivalent classes of line bundles?
In fact I am almost sure this is true (because they looks natural), but I do not know how to show this formally?