I am looking for examples of locally finitely presentable categories which admit a symmetric monoidal structure, such that the tensor product preserves colimits in each variable, but the unit is not finitely presentable, and/or there is a tensor product of finitely presentable objects which is not finitely presentable. 


Are there examples which appear in practice?


(The correct definition of a locally finitely presentable tensor category is one where the unit object is finitely presentable, and the tensor product of two finitely presentable objects is finitely presentable; see for instance this [this paper](http://archive.numdam.org/article/CTGDC_1982__23_1_3_0.pdf) by Kelly. But I wonder if this is automatic - probably not.)



locally finitely presentable tensor categories
One can take the category of modules over a Laurent polynomial ring in one variable $\textrm{Mod}\;k[t,t^{-1}]$ and think of $k[t,t^{-1}]$ as the group algebra of $\mathbb{Z}$. The corresponding cocommutative Hopf algebra structure provides a symmetric monoidal structure $(\otimes\_k, k)$ on $\textrm{Mod}\;k[t,t^{-1}]$ which certainly preserves colimits in each variable. The unit $k$ is finitely presented, but for instance
$$
k[t,t^{-1}] \otimes\_k k[t,t^{-1}]
$$
is a tensor product of finitely presented objects which is not finitely presented.


One gets plenty of other examples along these lines.


