Let $R$ be graded ring $M,N$ be graded $R$-modules. If $f:M \longrightarrow N$ is isomorphism of $R$-modules (NOT graded), is $f$ isomorphism of graded $R$-module? In other words, if $M,N$ are isomorphic as $R$-modules, does it implies they isomorphic as graded $R$-modules?
I guess no since we are not sure that $f$ is morphism in the category of graded modules. But I have no example!
This may depend on your ring. If $R$ is concentrated in degrees $\geq 0$, then a copy of $R$ with bottom class in degree $0$ is isomorphic to $R$ shifted so the bottom class is in degree $1$, as ungraded modules, but not via any map of degree zero. If $R = k[x^{\pm 1}$ with $x$ in degree 1, this type of example won’t work and maybe your question has a different answer.