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?
Edit: More specifically, let $Y$ be an open subset of $X$ and let $\mathfrak{U}$ be an open cover of $Y$. The cosheaf condition for $\mathcal{O}'_X$ says that $$ \mathcal{O}'_X(Y) \longleftarrow \bigoplus_U \mathcal{O}'_X(U) \overset{r'}{\longleftarrow} \bigoplus_{U,V} \mathcal{O}'_X(U\cap V) $$ is a cokernel sequence in the category of vector spaces. The maps are the duals of the maps from kernel sequence $$ \mathcal{O}_X(Y) \longrightarrow \prod_U \mathcal{O}_X(U) \overset{r}{\longrightarrow} \prod_{U,V} \mathcal{O}_X(U\cap V) $$ in locally convex vector spaces from the sheaf condition for $\mathcal{O}_X$ where $r$ sends $(f_U)_U$ to $(f_U|_{U\cap V} - f_V|_{U\cap V})_{U,V}$. Let $A$ and $B$ denote the source resp. the target of $r$. Let $R: A/\ker r \rightarrow \mathrm{im}\ r$ be the map induced by $r$ where $\mathrm{im}\ r$ has the subspace topology in $B$. The answer to the above question would be yes if the dual map $R'$ were surjective. This condition appears when trying to prove that the natural map $\mathrm{coker} (r') \rightarrow (\ker r)' = \mathcal{O}'_X(Y)$ is injective by using the Hahn-Banach theorem. Surjectivity follows from the Hahn-Banach theorem. In the case of $Y$ an open subset of $\mathbf{C}$ and a cover consisting of two open subsets $R'$ is an isomorphism because $R$ is a bijective continuous map between Fréchet spaces and the open mapping theorem applies.
