Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proof. If $\overline{f}: B^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram extending $f$, then Corollary 7.2.2.3 guarantees that $\overline{f} \circ e^{\triangleleft }: A^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram extending $f \circ e$. Conversely, if $f \circ e$ can be extended to a colimit diagram, then Proposition 7.2.2.9 (applied in the special case $\operatorname{\mathcal{D}}= \Delta ^0$) guarantees that $f$ can also be extended to a colimit diagram. $\square$