Consider $H \left(U \right); U \subset \mathbb{C}$ - space of holomorphic functions with compact-open topology. In this topology, this space is Montel, nuclear and Frechet. I want to take the inductive limit of tensor product of 2 copy of this space and I am wondering:

$$ \varinjlim\limits_{U \supset K} \left( H \left(U \right) \hat{\bigotimes} H \left(U \right) \right) \stackrel{?}{=} H \left(K \right) \hat{\bigotimes} H \left(K \right) = H \left(K \times K\right), $$ where $H \left( K \right)-$ space of germs of holomorphic functions on compact set $K \subset \mathbb{C}$. It doesn't matter which type of tensor product we take because space is nuclear.