Anodyne morphisms can be defined as the closure of horn inclusions $\Lambda^k_n\to \Delta^n$ along retracts, transfinite composition and pushouts. A nice result on these anodyne morphisms is that the pushout product map is again anodyne. That is, if $f:X\to Y$ and $i:A\to B$ are two monos, of which one of them is anodyne, then the map $$(A\times Y)\coprod_{X\times A}(B\times X)\to Y\times B$$ is anodyne.
From this lemma probably, it should be derivable from that the product of two anodyne maps is also anodyne. It suffices to show that for $f$ anodyne, the product $f\times id$ is anodyne, but I just fail to see this. Does someone have any idea how to do this?
Thanks!
edit: adding a diagram for the pushout product of $f$ with the identity. 
Which is the same as the following pushout product diagram
But this only says that the identity on $Y\times A$ is anodyne, which is not really a suprise. Probably I do really have some blind spot, or my category theory has gone too rusty.