Let \(T\colon\mathcal A\to\mathcal B\) be an additive covariant functor between abelian categories, such as categories of modules. The functor \(T\) is right exact if for every short exact sequence \(0\to A\mapsvia{f}B\mapsvia{g}C\to0\) in \(\mathcal A\) the sequence \[ T(A)\mapsvia{T(f)}T(B)\mapsvia{T(g)}T(C)\to0 \] is exact, and left exact if for every such sequence \[ 0\to T(A)\mapsvia{T(f)}T(B)\mapsvia{T(g)}T(C) \] is exact. It is exact if it is both left and right exact, that is, if it takes every short exact sequence to a short exact sequence.
For a commutative ring \(R\) and an \(R\)-module \(N\), the functor \(-\otimes_RN\) on \(R\)-modules is right exact [@DF04], and so is \(N\otimes_R-\).