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Let $\mathcal C,\mathcal D$ be two Abelian categories, and $T$ be a functor which satisfies

for any objects $A,B$ of $\mathcal C,\mathcal D$ respectively, $T$ induces an isomorphism $\operatorname{Hom} (A,B) \cong \operatorname{Hom} (TA,TB)$.

Then, is there a special name for such $T$?

Thanks to everyone.

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$T$ is called additive if the map $\mathrm{Hom}(A,B)\to \mathrm{Hom}(TA,TB)$ is a group homomorphism. $T$ is called fully faithful if the map $\mathrm{Hom}(A,B)\to \mathrm{Hom}(TA,TB)$ is a bijection. So I would call such a functor a fully faithful additive functor.

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  • $\begingroup$ Thank you very much for the answer. $\endgroup$ – IzumiEternal Feb 9 '17 at 7:24

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