what is a faithfully exact functor?

Could any of you give me a definition of faithfully exact functor, please?

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I think it's a functor which both preserves and reflects short exact sequences, but I can't find a reference which actually says this. –  Qiaochu Yuan Jul 4 '11 at 11:22

Indeed a functor $F:\mathcal{A}\to \mathcal{B}$ of abelian categories is called faithfully exact if the following holds: A sequence $A\to B\to C$ in $\mathcal{A}$ is exact if and only if the induced sequence $F(A)\to F(B)\to F(C)$ in $\mathcal{B}$ is exact.
The motivation for this terminology is probably the following: Let $R$ be a ring and $M$ an $R$-module. Then the functor $M\otimes_R -$ is exact if and only if $M$ is flat and it is faithfully exact if and only if $M$ is faithfully flat.