Let $\Lambda=\mathbb Z _p[[ T]]$ be the ring of power series over the $p$-adic integers, and let $X$ and $Y$ be $\Lambda$-modules. We say that $X$ is pseudo-isomorphic to $Y$ if there exists a $\Lambda$-module morphism $f:X\to Y$ with finite kernel and cokernel.
It is mentioned in the literature (e.g. Lang's Cyclotomic Fields) that pseudo-isomorphism is symmetric for torsion $\Lambda$-modules. Neither of the aforementioned books provide a proof. Is there something elementary I'm missing?
(Note that in general, pseudo-isomorphism is not symmetric: consider $X=(p,T)$ and $Y=\Lambda$. Then the inclusion $X\hookrightarrow Y$ is a pseudo-isomorphism, but every morphism $Y\to X$ has infinite cokernel.)