Kerodon

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

Remark 5.1.6.3. Condition $(B)$ of Definition 5.1.6.1 can be rephrased as follows: given any commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & \overline{Y} \ar [dr]^{ \overline{g} } & \\ \overline{X} \ar [rr]^{ \overline{h} } \ar [ur]^{ \overline{f} } & & \overline{Z} } \]

in the category $\operatorname{\mathcal{C}}$ and any partially defined lift

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar@ {-->}[ur]^{f} \ar [rr]^{h} & & Z } \]

to a diagram in $\operatorname{\mathcal{E}}$ (so that $U(g) = \overline{g}$ and $U(h) = \overline{h}$), there exists a unique extension as indicated (that is, a unique morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ satisfying $U(f) = \overline{f}$).