I was just introduced to the notion of projective module (through Matsumura, Commutative ring theory), and there it is said that the reason that there is no easy description for injective modules (along the lines of "projective modules are direct summands of free modules") is that there is no notion dual to free module. Could somebody enlighten me about the difficulty of such construction, i.e., why would it be meaningless to reverse the arrows in the universal property definition of free module?
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Caveat lector: this answers your last question "why would it be meaningless to reverse the arrows in the universal property definition of free module?", regardless of the fact that this is not exactly what you want for your purposes: see Qiaochu's comment below.
A cofree functor, the one that gives you cofree objects, i.e. those who satisfy the dual universal property to free objects, would be a right adjoint to the forgetful functor.
Alas, the forgetful functor does not have a right adjoint in the category of modules over a ring, because it would have to preserve colimits, and it doesn't (e.g. it doesn't preserve coproducts).
Hence there is no "cofree functor".