Whether you capitalize a dog's breed depends on the origin of the breed name.
In general, the rule for dog (and cat) breeds is to capitalize the part of the name derived from a proper noun and lowercase the part of the name derived from a common noun.
In the following examples, "English" and "Yorkshire" are derived from proper nouns—the names of places—and thus should be capitalized. On the other hand, "mastiff" is derived from the Latin "mansuétus" for "tame or mild," and "terrier" is derived from Old French "chien terrier" for "dog of the earth, land, or ground"—both common nouns, which means those parts should not be capitalized.
When in doubt, consult a dictionary.
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