Recall that $\operatorname{Sch}$ can be identified with the subcategory of (Zariski-)locally representable (by an affine) étale sheaves on $\operatorname{CRing^{op}}$. In this case, $\operatorname{Spec}(-):\operatorname{CRing^{op}}\to \operatorname{Sch}$ is simply the co-Yoneda embedding $R\mapsto \operatorname{Hom}(R,-)$ (which makes sense since the étale topology is subcanonical and $\operatorname{Hom}(R,-)$ is obviously locally affine).

Is there a nice "sheaf-theoretic" description of $\operatorname{Proj}:\operatorname{(Gr_{{\mathbf Z}_{\geq 0}}CRing)^{op}}\to \operatorname{Sch}$ (I've only seen $\operatorname{Proj}$ for nonnegative integral grading. If we can use more exotic gradings, I guess I'd be interested in that too). Hoping for it to be as nice as $\operatorname{Spec}$ seems like a bit of a pipe dream, but I'm wondering if there is a nicer way to describe it than Hartshorne's approach, which feels rather arbitrary.

Edit: To clarify, I'm looking for a construction $\operatorname{Proj}:\operatorname{(Gr_{{\mathbf Z}_{\geq 0}}CRing)^{op}}\to \operatorname{Sh}_{\acute{et}}(\operatorname{CRing^{op}})$, which we can then see lands in $\operatorname{Sch}$ by showing that we can cover it with Zariski-open affines.