I have a proper model category and in it two coequalisers, $A_i \rightrightarrows B_i \to C_i$, $i=1,2$. I have a map of diagrams arising from maps $A_1 \to A_2$, $B_1 \to B_2$ where these two arrows are cofibrations. Can I reasonably expect the canonical arrow $C_1 \to C_2$ to be a cofibration? Are there additional properties on the model category I can assume such that it is?
It may help that the maps $A_i \to B_i$ could also be cofibrations, if I am not mistaken.