 Morphism on Proj induced by a graded ring homomorphism

A graded homomorphism $S\to T$ induces a morphism to $\mathrm{Proj}S$ on the open subset of $\mathrm{Proj}T$ where the inverse image of a prime does not contain $S_{+}$. On $D_{+}(\phi (f))$ it is induced by $(S_f{)}_0\to (T_{\phi (f)}{)}_0$.