For $K\subseteq \mathbb{R}^d$ compact, let $C_{\mathrm{c}}^{\infty}(K)$ denote the space of smooth functions on (an open neighborhood of) $K$ with compact support contained in $K$ with the usual Fréchet space subspace topology inherited from $C^{\infty}(K)$, and denote by $C_{\mathrm{c}}^{\infty}(K)^*$ the dual with the Mackey topology (or, equivalently, the strong polar topology). For $U\subseteq \mathbb{R}^d$ open, let $C_{\mathrm{c}}^{\infty}(U)$ denote the LF-space of smooth functions with compact support contained in $U$, and denote by $C_{\mathrm{c}}^{\infty}(U)^*$ the space of generalized-functions on $U$, again, with the Mackey topology (or equivalently the strong polar topology).

If $K\subseteq U$, we have extension-by-zero maps $C_{\mathrm{c}}^{\infty}(K)\rightarrow C_{\mathrm{c}}^{\infty}(U)$, and we thereby obtain restriction maps $C_{\mathrm{c}}^{\infty}(U)^*\rightarrow C_{\mathrm{c}}^{\infty}(K)^*$.

Is it true that $\operatorname{colim}_{\substack{U\supseteq K \\ U\text{ open}}}C_{\mathrm{c}}^{\infty}(U)^*=C_{\mathrm{c}}^{\infty}(K)^*$? That is, is $C_{\mathrm{c}}^{\infty}(K)^*$ the colimit in the category of locally-convex spaces of $C_{\mathrm{c}}^{\infty}(U)^*$ as $U$ ranges over all open sets which contain $K$?

For what it's worth, the motivation is that this colimit is the usual definition of the sheaf on a not-necessarily-open subset $K$ (or if you like, the global sections of the inverse image sheaf under the inclusion $K\hookrightarrow \mathbb{R}^d$), and so it would be nice if this colimit had a relatively concrete description.