If $M$ is a left $\mathbb{C}[t] \langle \partial _t \rangle$-module ( a left module over the Weyl algebra), then $\mathrm{Hom}(M,\mathbb{C}[t] \langle \partial _t \rangle)$ is equipped with a structure of right $\mathbb{C}[t] \langle \partial _t \rangle$-module (by $(\varphi . P)(m) = \varphi(m)P$). If we replace the Weyl algebra by a ring of differential operators $k\langle \partial _t \rangle$ (when $k$ is a (skew)field) and consider a left module $N$ then, Is $\mathrm{Hom}(N,k\langle \partial _t \rangle)$ a right module with the action defined above? In what other contexts this "duality" left-right could be considered?

Thanks in advance