This is a quick question from Frohlich and Taylor's Algebraic Number Theory, II.4, p 94.
Let $R$ be a Dedekind domain with quotient field $K$, $\mathfrak p$ is a non-zero prime ideal of $R$ and $R_\mathfrak p$ the completion of $R$ wrt the valuation associated to $\mathfrak p$.
$L$ and $M$ are $R$-lattices in a finite dimensional vector space $V$ (finitely generated, torsion-free $R$-modules spanning $V$ over $K$). Let $L_\mathfrak p$ be the $R_\mathfrak p$-module $L\otimes_R R_\mathfrak p$ and similarly for $M_\mathfrak p$ and $V_\mathfrak p\cong V\otimes_K K_\mathfrak p$.
The book says there is an automorphism $l_\mathfrak p$ of $V_\mathfrak p$ with $l_\mathfrak p(L_\mathfrak p)=M_\mathfrak p$ and that it's unique modulo ${\rm Aut}_{R_\mathfrak p}(M_\mathfrak p)$.
Why is the last bit true? I am confused about what freedom there is in the choice of $l_\mathfrak p$.
Thanking you for your help!