Remark 35.4.11. We will use frequently the standard adjunction between $\mathop{\mathrm{Hom}}\nolimits $ and tensor product, in the form of the natural isomorphism of contravariant functors

taking $f: M_1 \otimes _ R M_2 \to \mathbf{Q}/\mathbf{Z}$ to the map $m_1 \mapsto (m_2 \mapsto f(m_1 \otimes m_2))$. See Algebra, Lemma 10.14.5. A corollary of this observation is that if

is a split coequalizer diagram in $\text{Mod}_ R$, then so is

for any $Q \in \text{Mod}_ R$.

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