See here

For each object in a category, we can define a representable functor which is a Functor from a category to Set, where objects are mapped to Hom-sets to (or from) that object, and morphisms are mapped to morphisms between arrows defined by composition. There is a covariant and a contravariant version.

There is another functor, from a category to {the Category of functors from the category to Set} where objects are mapped to to their corresponding Representable functors, and morphisms are Natural transformations induced by morphisms in the original category. This is called the Yoneda embedding.

where does the name "representable" come from?