Definition 1.5.4.1. Let $\operatorname{\mathcal{C}}$ be a category. A lifting problem in $\operatorname{\mathcal{C}}$ is a commutative diagram $\sigma :$
\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r]^-{u} & X \ar [d]^{g} \\ B \ar [r]^-{v} & Y } \]
in $\operatorname{\mathcal{C}}$. A solution to the lifting problem $\sigma $ is a morphism $h: B \rightarrow X$ in $\operatorname{\mathcal{C}}$ satisfying $g \circ h = v$ and $h \circ f = u$, as indicated in the diagram
\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r]^-{u} & X \ar [d]^{g} \\ B \ar [r]^-{v} \ar [ur]^{h} & Y. } \]