Let $X$ be a complex-analytic manifold. Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we give $\mathcal{O}_X(U)$ the topology of uniform convergence on compact subsets of $U$. The precosheaf $\mathcal{O}'_X$ of analytic functionals is defined by taking the continuous dual of $\mathcal{O}_X$, that is, $\mathcal{O}'_X(U) := \mathcal{O}_X(U)'$ where the latter denotes the vector space of continuous functionals.

Is $\mathcal{O}'_X$ a cosheaf of vector spaces?