Proof.
Unwinding the definitions, we see that $\boldsymbol {\sqcup }^{I}_{i}$ can be identified with the pushout
\[ (\{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ) \coprod _{ \{ 1\} \times \operatorname{\partial \raise {0.1ex}{\square }}^{I \setminus \{ i\} } } (\Delta ^1 \times \operatorname{\partial \raise {0.1ex}{\square }}^{I \setminus \{ i\} } ). \]
Consequently, a map of simplicial sets $f: \boldsymbol {\sqcup }^{I}_{i} \rightarrow X_{\bullet }$ can be identified with a commutative diagram of solid arrows
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \raise {0.1ex}{\square }}^{I \setminus \{ i\} } \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^1, X_{\bullet } ) \ar [d] \\ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \ar [r] \ar@ {-->}[ur] & \operatorname{Fun}( \{ 1\} , X_{\bullet } ), } \]
and an extension $\overline{f}: \operatorname{\raise {0.1ex}{\square }}^{I} \rightarrow X_{\bullet }$ of $f$ can be identified with a solution to the associated lifting problem. If $X_{\bullet }$ is a Kan complex, then the right vertical arrow is a trivial Kan fibration (Theorem 2.4.5.11), so the desired extension exists by virtue of Proposition 1.5.5.4. This proves $(a)$; the proof of $(b)$ is similar.
$\square$