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
$$\label{descent-equation-adjunction} C(\bullet _1 \otimes _ R \bullet _2) \cong \mathop{\mathrm{Hom}}\nolimits _ R(\bullet _1, C(\bullet _2)): \text{Mod}_ R \times \text{Mod}_ R \to \text{Mod}_ R$$

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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