Let $$\begin{array}{ccccccccc} A & \rightarrow & X \\ i\downarrow & & \downarrow p \\ B & \xrightarrow{v} & S \end{array} $$ be a commutative diagram of simplicial sets, with $p$ an inner fibration. Two solutions $f,g$ to this lifting problems are called homotopic relative to $A$ over $S$ provided that they are equivalent in the fiber of the map $$X^B\rightarrow X^A\times_{S^A}S^B.$$

Now, it is easy to see that this imply the existence of a map $\Delta^1\times X\rightarrow X$ satisfying certain obvious properties (spelled out, for instance, here: https://ncatlab.org/nlab/show/minimal+inner+fibration), amongst which is that the map $$F|_{\{b\}\times\Delta^1}\rightarrow X_{v(b)}$$ is an equivalence for every $b$.

How can you prove the converse? I found an argument for the case when $A=\emptyset$ and $S=*$, but not for the general case.

NB: Chances are it is proven at some point in HTT, but so far, I have only found this assertion without proof.

Let $$\begin{array}{ccccccccc} A & \rightarrow & X \\ i\downarrow & & \downarrow p \\ B & \xrightarrow{v} & S \end{array} $$ be a commutative diagram of simplicial sets, with $p$ an inner fibration. Two solutions $f,g$ to this lifting problems are called homotopic relative to $A$ over $S$ provided that they are equivalent in the fiber of the map $$X^B\rightarrow X^A\times_{S^A}S^B.$$

Now, it is easy to see that this imply the existence of a map $\Delta^1\times B\rightarrow X$ satisfying certain obvious properties (spelled out, for instance, here: https://ncatlab.org/nlab/show/minimal+inner+fibration), amongst which is that the map $$F|_{\{b\}\times\Delta^1}\rightarrow X_{v(b)}$$ is an equivalence for every $b$.

How can you prove the converse? I found an argument for the case when $A=\emptyset$ and $S=*$, but not for the general case.

NB: Chances are it is proven at some point in HTT, but so far, I have only found this assertion without proof.

Let $$\begin{array}{ccccccccc} A & \rightarrow & X \\ i\downarrow & & \downarrow p \\ B & \xrightarrow{v} & S \end{array} $$ be a commutative diagram of simplicial sets, with $p$ an inner fibration. Two solutions $f,g$ to this lifting problems are called homotopic relative to $A$ over $S$ provided that they are equivalent in the fiber of the map $$X^B\rightarrow X^A\times_{S^A}S^B.$$

Now, it is easy to see that this imply the existence of a map $\Delta^1\times X\rightarrow X$ satisfying certain obvious properties (spelled out, for instance, here: https://ncatlab.org/nlab/show/minimal+inner+fibration), amongst which is that the map $$F|_{\{b\}\times\Delta^1}\rightarrow X_{v(b)}$$ is an equivalence for every $b$.

How can you prove the converse?

NB: Chances are it is proven at some point in HTT, but so far, I have only found this assertion without proof.

Let $$\begin{array}{ccccccccc} A & \rightarrow & X \\ i\downarrow & & \downarrow p \\ B & \xrightarrow{v} & S \end{array} $$ be a commutative diagram of simplicial sets, with $p$ an inner fibration. Two solutions $f,g$ to this lifting problems are called homotopic relative to $A$ over $S$ provided that they are equivalent in the fiber of the map $$X^B\rightarrow X^A\times_{S^A}S^B.$$

Now, it is easy to see that this imply the existence of a map $\Delta^1\times X\rightarrow X$ satisfying certain obvious properties (spelled out, for instance, here: https://ncatlab.org/nlab/show/minimal+inner+fibration), amongst which is that the map $$F|_{\{b\}\times\Delta^1}\rightarrow X_{v(b)}$$ is an equivalence for every $b$.

How can you prove the converse? I found an argument for the case when $A=\emptyset$ and $S=*$, but not for the general case.

NB: Chances are it is proven at some point in HTT, but so far, I have only found this assertion without proof.