Let $X$ be a scheme and $\mathcal{A}$ be a sheaf of $\mathbb{Z}_{\ge 0}$-graded $\mathcal{O}_X$-algebras.
From the data above there is a construction which gives the "global Proj" $Proj \mathcal{ A} \to X$ which satisfies that the pullback along any affine open subset $SpecB=U \subset X$ is the (usual) Proj construction of the graded $B$-algebra $\Gamma(U,\mathcal{A})$.
I've went through several proofs of this construction over the past few months and was rather patient hoping that eventually I'll be convinced. Sadly though I'm still confused just as badly as I was on day 1. My problem is that all the ad-hoc arguments about gluing don't stick with me in the long run since I'm not entirely sure what object is being constructed and what properties are being verified. For the main example of what my problem is look at the following question
Suppose I choose an affine cover (assuming $X$ is qcqs) and construct $Proj \mathcal{A}$ via a gluing. Suppose now I choose a different cover, what garauntees me that the result will be the same? Moreover why do pullbacks behave nicely as in the above description?
Here is an attempt of mine to give a universal construction which might solve the issue:
The following limit (assuming it exists) seems natural from the point of view of gluing a space over a basis of open sets (the limit is indexed over the category of coverings with refinements as morphisms):
$$\lim_{\text{all covers}} \text{coeq} \{ \coprod Proj \Gamma(U_{ij}, \mathcal{A})\rightrightarrows \coprod Proj \Gamma(U_{j}, \mathcal{A}) \}$$
Is this limit isomorphic to the global Proj of $\mathcal{A}$? Perhaps after imposing some reasonable conditions on $X$?