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Suppose $\mathcal{A,B,C}$ are additive categories, $\mathcal B$ is a subcategory of $\mathcal C$. Now let $F,G: \mathcal A\rightarrow\mathcal B$ be two additive functors. Suppose $F,G$ are naturally equivalent as functors $\mathcal{A}\rightarrow\mathcal C$, do they induce the same map $K_n(\mathcal A)\rightarrow K_n(\mathcal B)$? Of course, $\mathcal{A,B,C}$ are endowed with the split exact structures. Thanks!

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Yes if the natural isomorphisms $F(X)\cong G(X)$ are in $\mathcal B$. –  Fernando Muro Apr 16 '13 at 7:57
Thanks Fernando, that's right if the natural isomorphisms are in $\mathcal B$ by applying the additivity theorem. I don't know whether it is still true if the isomorphisms are in the ambient category $\mathcal C$. I guess probably it's not true. –  Kun Wang Apr 16 '13 at 13:02

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