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His mother is heterozygous for the E gene which controls whether a horse is black or red based. So she can pass along an E allele or an e allele.

His father is homozygous for the recessive e allele. So he can only pass on an e allele.

Horses that are Ee or EE are black based and horses that are ee are red based. So your colt got a e from mom and a e from Dad. Resulting in the red or chestnut base color.

Take a peek at the gene here:
http://www.huntandjump.com/geneticsguide/?page_id=27

Peaches is Ee. Mission has to be ee to be chestnut anyway. Peaches passed her e, so the colt is ee and therefore chestnut/red instead of bay. Peaches is homozygous roan, so all her foals will be roan. He's got a copy of sooty from his dad but Mission must be heterozygous DP and the colt missed out on that part....