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.



Cofibrations and coequalisers in a proper model category
No. Think of examples in which $A\_1$ is initial (so $B\_1=C\_1$). There's no reason why $B\_1\to C\_2$ should be a cofibration just because $B\_1\to B\_2$ is.


