Warning 8.5.1.21. If $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, then a retraction diagram in $\operatorname{\mathcal{C}}$ can be identified with a pair of morphisms $i: Y \rightarrow X$ and $r: X \rightarrow Y$ satisfying the condition $r \circ i = \operatorname{id}_{Y}$. Beware that, if $\operatorname{\mathcal{C}}$ is a general $\infty $-category, then a retraction diagram
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{ r } & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_ Y } & & Y } \]
generally cannot be recovered (even up to isomorphism) from the morphisms $i$ and $r$ alone: one also needs a homotopy which witnesses the identity $[r] \circ [i] = [ \operatorname{id}_{Y} ]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.