Answer:
You can assign any type of letter to the characteristic. It is common to use the first letter of the dominant trait capitalized and the lower case for the recessive trait in this case.
Dominant is black hair (B)
Recessive trait is chestnut color (b).
Being homozygous means having 2 alleles of the same kind (so you could have dominant homozygous or recessive homozygous.. Being heterozygous means that you have one allele dominant and one recessive.
If the black horse is homozygous, the genotype is BB.
If the black horse is heterozygous, the genotype is Bb. Where Black is dominant and is expressed over the chestnut color.