[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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