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This has some overlap with the existing entry functorial geometry.
Yes, I will add a link. We need again both entries as the functorial geometry is a general approach to spaces due Grothendieck, while the fact that schemes (as locally ringed spaces) are equivalent to a category of spaces on $Aff$ in Zariski Grothendieck topology is a nontrivial theorem which deserves its own entry (if schemes are defined in a standard way). The fundamental theorem of morphisms of schemes is just a part of that theorem which claims embedding into a presheaves, the rest is about characterizing the essential image via local representability + sheaf condition. A similar statement (with another topology) should be somewhere stated for algebraic spaces.
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