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I try to understand the Grothendieck philosophy. In the MochizukiĀ“s paper Categorical Representation of Local Noetherian Log Schemes, Mochizuki characterizes an algebraic variety using the subcategory of the finite type morphisms in the comma category. I want know if it result can be generalizated for a arbitrary category, using the comma category and as Theo Johnson-Freyd notice the forgetfull funtor.

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    $\begingroup$ Your intuition cannot be quite correct. What happens when the ambient category consists of two points with no nontrivial morphisms? I think a correct version of Yoneda says that an object $X$ in a category $\mathcal C$ can be recovered from the comma category $\mathcal C_{/X}$ together with the forgetful functor $\mathcal C_{/X} \to \mathcal C$. $\endgroup$ – Theo Johnson-Freyd Jun 5 '17 at 21:05
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    $\begingroup$ The published version of the paper is freely available at doi.org/10.1016/j.aim.2003.10.005 $\endgroup$ – Philippe Gaucher Jun 6 '17 at 8:05