Proof.
By virtue of Corollary 4.4.3.15, condition $(1)$ is satisfied if and only if there exists a morphism of simplicial sets $u: Q \rightarrow \operatorname{Fun}_{A/\, /S}( B, X)$, where $Q$ is a contractible Kan complex, and a pair of vertices $x_0, x_1 \in Q$ satisfying $u(x_0) = f_0$ and $u(x_1) =f_1$. Moreover, we may assume (modifying $Q$ if necessary) that the vertices $x_0$ and $x_1$ are distinct. In this case, let $\overline{B} = A \coprod _{ (Q \times A) } (Q \times B)$ and let
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
be the categorical mapping cylinder described in Example 4.6.3.6. Unwinding the definitions, we see that morphisms $u: Q \rightarrow \operatorname{Fun}_{A/\, /S}( B, X)$ satisfying $u(x_0) = f_0$ and $u(x_1) = f_1$ can be identified with solutions to the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur] & S. } \]
This proves that $(3) \Rightarrow (1) \Rightarrow (2)$.
We will complete the proof by showing that $(2) \Rightarrow (3)$. Assume that $(2)$ is satisfied, so that the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ factors as a composition
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
where $\pi $ is a categorical equivalence and there exists a morphism $\overline{f}: \overline{B} \rightarrow X$ for which the diagram
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar [ur]^{ \overline{f} } & S } \]
commutes. Using Exercise 3.1.7.11, we can factor $\pi $ as a composition
\[ \overline{B} \xrightarrow {j} \overline{B}' \xrightarrow {\pi '} B, \]
where $j$ is a monomorphism and $\pi '$ is a trivial Kan fibration. Then $\pi '$ is also a categorical equivalence (Proposition 4.5.3.11), so the morphism $j$ is a categorical equivalence (Remark 4.5.3.5). Our assumption that $q$ is an isofibration guarantees that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \overline{B} \ar [d]_{j} \ar [r]^-{ \overline{f} } & X \ar [d]^{q} \\ \overline{B}' \ar [r]^-{ g \circ \pi ' } \ar@ {-->}[ur]^{ \overline{f}' } & S } \]
admits a solution $\overline{f}': \overline{B}' \rightarrow X$.
We now show that condition $(3)$ is satisfied. Suppose that we are given another factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): (\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition
\[ B \coprod _{A} B \xrightarrow {\iota } \overline{B}'' \xrightarrow {\pi ''} B, \]
where $\iota $ is a monomorphism. We wish to show that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ \iota } \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B}'' \ar [r]^-{ g \circ \pi ''} \ar@ {-->}[ur]^{\overline{f}''} & S } \]
admits a solution $\overline{f}'': \overline{B}'' \rightarrow X$. We first observe that the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [r]^-{j \circ (s_0, s_1)} \ar [d]_{\iota } & \overline{B}' \ar [d]^{ \pi ' } \\ \overline{B}'' \ar [r]^-{\pi ''} \ar@ {-->}[ur]^{v} & B } \]
admits a solution $v: \overline{B}'' \rightarrow \overline{B}'$, since $\iota $ is a monomorphism and $\pi '$ is a trivial Kan fibration. We now conclude the proof by setting $\overline{f}'' = \overline{f}' \circ v$.
$\square$