Kerodon

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

Remark 4.2.5.3. Condition $(B)$ of Definition 4.2.5.1 can be rephrased as follows: given any commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & \overline{Y} \ar [dr]^{ \overline{v} } & \\ \overline{X} \ar [rr]^{ \overline{w} } \ar [ur]^{ \overline{u} } & & \overline{Z} } \]

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

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{v} & \\ X \ar@ {-->}[ur]^{u} \ar [rr]^{w} & & Z } \]

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