I have some questions about how to apply proposition 2.4.1.8 in HTT which says: In several places, variant forms of this proposition have been used without explanation such as in the proof of Lemma 2.3.3.5: My question is: In 2.4.1.8 we are dealing with the smash product of $\{0\}\rightarrow \Delta^1$ and $\partial \Delta^n\rightarrow \Delta^n$, but in 2.3.3.5 the morphism concerned is the smash product of $\Lambda^2_2\rightarrow \Delta^2$ and $\partial \Delta^n\rightarrow \Delta^n$, How to reduce the case in 2.3.3.5 to the case in 2.4.1.8?

My attempt:we can use 2.4.1.8 to extend $(\Delta^2\times \partial\Delta^n)\coprod_{\Lambda^2_2\times \partial \Delta^n}(\Lambda^2_2\times \Delta^n)\rightarrow C$ as $\partial \Delta^2\times\Delta^1\rightarrow C$.But how can I deal with the interior of $\Delta^2$?