I recently read that a cofree $R$-module is defined in such a way that some properties of free modules and some properties of cofree modules are dual. For example see here.
Every free module is projective; every cofree module is injective.
For every module $M$, there is a surjective homomorphism from a free module to $M$; for every module $M$, there is an injective homomorphism from $M$ to a cofree module.
A module is projective iff it can be completed by a direct sum to a free module; a module is injective iff it can be completed by a direct product to a cofree module.
But it is also pointed that free and cofree modules are not exactly duals of each other in some sense.
I was thinking that there could be a property of free modules, whose dual does not hold for cofree modules and vice-versa.
Question: What are simple properties of free modules whose duals do not hold true for co-free modules? (Similar question with free and cofree interchanged!)