Is the inclusion of the wedge sum into the reduced cylinder a relative cell complex?

For a CW-complex $$X$$ the map $$X\coprod X \hookrightarrow X \times I$$ is a relative cell complex. What I want to know is if this still holds in the pointed case. That is, if $$X$$ is a pointed CW-complex with non-degenerated base point and $$I_+ = [0,1] \coprod *$$, then is it true that $$X\vee X \hookrightarrow X \wedge I_+$$ is a relative cell complex?

Yes, it is a CW pair since $$X \wedge I_+$$ is the quotient of $$X \times I$$ by $$\{*\} \times I$$. It is easy to show that any point in a CW complex can be a 0-cell of a CW structure on it, so this quotient has a CW structure, and the image of $$X \sqcup X$$ under the quotient map is the subcomplex $$X \vee X$$.