left inverse to trivial fibration is trivial cofibration

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It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration.

Now, I see that there is a section, by using lifting property with $\emptyset \to Y$ on the left (all objects are cofibrant!). I see from 2-out-of-3 that this section is an equivalence. How can I see that it is a cofibration?

Thank you, Sasha