Kerodon

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Exercise 8.5.0.3. Let $\operatorname{\mathcal{C}}$ be a category containing a retraction diagram

\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y. } \]

Show that there is a unique functor $F: \operatorname{Ret}\rightarrow \operatorname{\mathcal{C}}$ which is given on objects by $F( \widetilde{X} ) = X$ and $F( \widetilde{Y} ) = Y$, and also satisfies $F( \widetilde{i} ) = i$ and $F( \widetilde{r} ) = r$. We therefore obtain a bijection

\[ \{ \textnormal{Functors $\operatorname{Ret}\rightarrow \operatorname{\mathcal{C}}$} \} \xrightarrow {\sim } \{ \textnormal{Retraction diagrams in $\operatorname{\mathcal{C}}$} \} . \]

In particular, an object $Y \in \operatorname{\mathcal{C}}$ is a retract of another object $X \in \operatorname{\mathcal{C}}$ if and only if there exists a functor $F: \operatorname{Ret}\rightarrow \operatorname{\mathcal{C}}$ satisfying $F( \widetilde{X} ) = X$ and $F( \widetilde{Y} ) = Y$.