Remark 4.2.6.3 (The Homotopy Extension Lifting Property). Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Unwinding the definitions, we see that the following conditions are equivalent:
The morphism $f$ is a left fibration.
For every monomorphism of simplicial sets $i: A_{} \hookrightarrow B_{}$, every lifting problem
\[ \xymatrix@C =100pt{ A_{} \ar [d]^{i} \ar [r] & \operatorname{Fun}( \Delta ^1, X_{} ) \ar [d]^{\operatorname{ev}_0} \\ B_{} \ar [r] \ar@ {-->}[ur] & \operatorname{Fun}( \{ 0\} , X_{} ) \times _{ \operatorname{Fun}( \{ 0\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) } \]admits a solution (indicated by the dotted arrow in the diagram).
For every monomorphism of simplicial sets $i: A_{} \hookrightarrow B_{}$, every lifting problem
\[ \xymatrix@C =100pt{ ( \Delta ^1 \times A_{} ) \coprod _{ \{ 0\} \times A_{} } (\{ 0\} \times B_{} ) \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^1 \times B_{} \ar [r]^-{\overline{h}} \ar@ {-->}[ur]^-{h} & S_{} } \]admits a solution (indicated by the dotted arrow in the diagram).
Let $u: B_{} \rightarrow X_{}$ be a map of simplicial sets and let $\overline{h}: \Delta ^1 \times B_{} \rightarrow S_{}$ be a map satisfying $\overline{h}|_{ \{ 0\} \times B_{} } = f \circ u$: that is, $\overline{h}$ is a homotopy from $f \circ u$ to another map $\overline{v} = \overline{h}|_{ \{ 1\} \times B_{} }$. Then we can choose a map of simplicial sets $h: \Delta ^1 \times B_{} \rightarrow X_{}$ satisfying $f \circ h = \overline{h}$ and $h|_{ \{ 0\} \times B_{} } = u$: in other words, $\overline{h}$ can be lifted to a homotopy $h$ from $u$ to another map $v = h|_{ \{ 1\} \times B_{} }$. Moreover, given any simplicial subset $A_{} \subseteq B_{}$ and any map $h_0: \Delta ^1 \times A_{} \rightarrow X_{}$ satisfying $f \circ h_0 = \overline{h}|_{ \Delta ^1 \times A_{}}$ and $h_0|_{ \{ 0\} \times A_{} } = u|_{ A_{}}$, we can arrange that $h$ is an extension of $h_0$.
In the special case where $B_{} = \Delta ^0$ and $A_{} = \emptyset $, each of these assertions reduces to the left path lifting property of $f$.