TITLE: Is there an extremal epi which is not epi, although binary products exist?
QUESTION [7 upvotes]: It is known, that in a category $\mathcal{A}$:

strong epis are epi, if $\mathcal{A}$ has binary products
extremal epis are epi, if $\mathcal{A}$ has equalizers
all strong epis are extremal 
all extremal epis are strong, if $\mathcal{A}$ has pullbacks

(of course, here I do not require epicness in the definition of strong or extremal epis)
Can it be that $\mathcal{A}$ has binary products and an extremal epi which is not epi? (bearing in mind that in this case $\mathcal{A}$ cannot have all pullbacks).

REPLY [5 votes]: How about the category of sets that are not singletons? Monomorphisms, epimorphisms and binary products are easily seen to be the same as in the category of all sets, but the map $f:\{x,y\}\to\{a,b\}$ with $f(x)=a=f(y)$ is extremal "epi".