The Stacks project

Remark 35.4.13. Let $f: M \to N$ be a universally injective morphism in $\text{Mod}_ R$. By choosing a splitting $g$ of $C(f)$, we may construct a functorial splitting of $C(1_ P \otimes f)$ for each $P \in \text{Mod}_ R$. Namely, by (35.4.11.1) this amounts to splitting $\mathop{\mathrm{Hom}}\nolimits _ R(P, C(f))$ functorially in $P$, and this is achieved by the map $g \circ \bullet $.