# The Stacks Project

## Tag 08WV

Remark 34.4.13. 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 (34.4.11.1) this amounts to splitting $\mathop{\rm Hom}\nolimits_R(P, C(f))$ functorially in $P$, and this is achieved by the map $g \circ \bullet$.

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1023–1032 (see updates for more information).

\begin{remark}
\label{remark-functorial-splitting}
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 (\ref{equation-adjunction}) this amounts to splitting $\Hom_R(P, C(f))$  functorially in $P$,
and this is achieved by the map $g \circ \bullet$.
\end{remark}

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