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[lemma 1.1] Let $\mathcal{C}$ be a preadditive category. Suppose G is in ($\mathcal{C^{op}}$, $\mathcal{Ab}$). If for each $X$ in $\mathcal{C}$ we are given a subgroup $A_x$ of $G(X)$ such that $G(f)(A_y)$ is contained in $A_X$ for all morphisms $f$ from $X$ to $Y$ in $\mathcal{C}$, then there is a unique subfunctor $F$ of $G$ such that $F(X)=A_x$ for all X in C.

I think the unique $F$ is the image of the natural transformation $Nat((-,X),G)$ in the Yoneda Lemma, but don't know how to prove it.

Here $A_X$ is "maximal" in a sense, and the uniqueness just reminded me the uniqueness in the Yoneda lemma.

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  • $\begingroup$ What is your question? How to prove the statement, or whether $F$ is given as you say? The statement is a straightforward consequence of the definitions (subfunctors are determined by the images of objects, and the existence is easy to check follows from the compatibility requirements given). $\endgroup$ Mar 9, 2015 at 9:04
  • $\begingroup$ Cross-posted to MSE. $\endgroup$ Mar 9, 2015 at 9:29
  • $\begingroup$ Thank for your comment ,I realised that the definition of F is just given in this way and need not to be proved, and the uniqueness then due to the "maximality" of Ax by checking the natural transformation between F and another F' which also satisfies F'(X)=Ax. $\endgroup$
    – milanelo
    Mar 9, 2015 at 11:57

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