Let $\Bbbk$ be a field. Let $S,T$ be graded $\Bbbk$-algebras, and let $\phi:S\to T$ be a graded ring map. We can assume finite generation if things become simpler. We have that if for sufficently large $d\gg0$, the degree $d$ part of the ring map $\phi_d:S_d\to T_d$ is an isomorphism, then $\phi$ induces $\mathrm{Proj}(T)\cong \mathrm{Proj}(S)$. My question is about the converse.
If $\phi:S\to T$ induces an isomorphism $\mathrm{Proj}(T)\cong \mathrm{Proj}(S)$, then do we have $\phi_d:S_d\cong T_d$ for large $d$?
According to the example given by @KReiser, I'd like to change the statement to
If $\phi:S\to T$ induces an isomorphism $\mathrm{Proj}(T)\cong\mathrm{Proj}(S)$, then do we have $\phi_d:S_d\cong T_d$ for sufficiently divisible $d\in\mathbb N_+$?