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
35.4.11.1
\begin{equation} \label{descent-equation-adjunction} C(\bullet _1 \otimes _ R \bullet _2) \cong \mathop{\mathrm{Hom}}\nolimits _ R(\bullet _1, C(\bullet _2)): \text{Mod}_ R \times \text{Mod}_ R \to \text{Mod}_ R \end{equation}
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
\[ \xymatrix@C=9pc{ C(M) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N) \ar[r] & C(P) } \]
is a split coequalizer diagram in $\text{Mod}_ R$, then so is
\[ \xymatrix@C=9pc{ C(M \otimes _ R Q) \ar@<1ex>[r] \ar@<-1ex>[r] & C(N \otimes _ R Q) \ar[r] & C(P \otimes _ R Q) } \]
for any $Q \in \text{Mod}_ R$.
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