Schemes of Objects in Abelian Categories

In the article Categorical Construction of Schemes, arXiv:2511.03433 we gave a natural definition of ordinary schemes based on the fact that the localization of a ring in a maximal ideal is a local representation of the corresponding function field. In this text, we replace the category of rings with a general locally small category $\text{cat}C$, we consider a subcategory $\text{cat}B\subset C$ of base-points, and assume that each $X\in \text{ob}\text{cat}C$ that contains $P\in \text{ob}\text{cat}B,$ i.e. there is a morphism $P\to X,$ there exists a local representing object $X_P\mathrm{.}$ Assuming that coproducts exists, we can use the construction of ordinary schemes to construct schemes of objects in any such category.