A module $M$ over a general ring R is called reflexive in case the canonical map $e_M: M \rightarrow M^{**}$ is an isomorphism, when $(-)^{*}:=Hom_A(-,A)$. Is $M$ reflexive iff $M \cong M^{**}$ for some isomorphism? I am interested in counterexamples or proofs for finitely generated or finitely presented $M$. See Are there non-reflexive modules isomorphic to their bi-dual? for a counter example for infinitely generated $M$. For a positive answer in special cases see Characterisation of reflexive modules .