Induced map in model categories

Question

In the snippet below I do not understand what is $$X_i\to X_i \coprod_{L_i X}L_i Y ,$$ i.e. how is it defined and why is it a cofibration.

Answer 1

The morphism $X_i\to X_i\coprod_{L_iX}L_iY$ is just the lower map of the pushout square $\require{AMScd}$ \begin{CD} L_iX @>>> L_iY \\ @VVV @VVV \\ X_i @>>> X_i\coprod_{L_iX}L_iY \end{CD}

Since $L_iX\to L_iY$ is a cofibration, so is its pushout along $L_iX\to X_i$ (as cofibrations are preserved by pushout); i.e., $X_i\to X_i\coprod_{L_iX}L_iY$ is a cofibration.