For every homogeneous $s\in S_+$,
$$
D_+(\varphi(s))
=\{\mathfrak p\in\operatorname{Proj}T:\varphi(s)\notin\mathfrak p\}.
$$
Hence
$$
U=\bigcup_{s\in S_+\text{ homogeneous}}D_+(\varphi(s)),
$$
so $U$ is open in the <Zariski topology>.

If $\mathfrak p\in U$, then $\varphi^{-1}(\mathfrak p)$ is a homogeneous <prime ideal> that does not contain the <irrelevant ideal of a graded ring> $S_+$. Thus
$$
f(\mathfrak p)=\varphi^{-1}(\mathfrak p)
$$
defines a map $U\to\operatorname{Proj}S$. On every <Standard affine open of Proj> the graded homomorphism induces
$$
(S_s)_0\longrightarrow (T_{\varphi(s)})_0,
$$
and therefore an <affine scheme> morphism
$$
D_+(\varphi(s))\longrightarrow D_+(s).
$$
These morphisms agree after localization on overlaps, so they glue to the required <morphism of schemes> $f:U\to\operatorname{Proj}S$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 113 / 1 / a / Solution

{title2=$S_+$}

For a nonnegatively graded ring $S=\bigoplus_{d\geq0}S_d$, the irrelevant ideal is
$$
S_+=\bigoplus_{d>0}S_d.
$$
The points of $\operatorname{Proj}S$ are precisely the homogeneous <prime ideals> that do not contain $S_+$.

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 Incoming links: Irrelevant ideal of a graded ring