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 (do you agree?). But I wonder if this is automatic - probably not.)

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 by Kelly. But I wonder if this is automatic - probably not.)