representable functor

{{category theory}} Given a category ''C'', a '''covariant representable functor''' ''h''^A sends each object ''X'' in category ''C'' to ''C''(''A'',''X''), the set of arrows from ''A'' to ''X'', and sends each arrow ''f'':''X'' &rarr; ''Y'' to a function from ''C''(''A'',''X'') to ''C''(''A'',''Y'') which maps each element ''s'' of ''C''(''A'',''X'') to an element <math> f \circ s</math> of ''C''(''A'',''Y''). A '''contravariant representable functor''' ''h''_''A'' sends each object ''X'' in category ''C'' to set ''C''(''X'',''A'') and each arrow ''f'':''X'' &rarr; ''Y'' to a function from ''C''(''Y'',''A'') to ''C''(''X'',''A'') which maps each element ''s'' of ''C''(''Y'',''A'') to an element <math>s\circ f</math> of ''C''(''X'',''A''). Note: ''h''^''A'':''C'' &rarr; '''Sets''' and ''h''_''A'':''C''^op &rarr; '''Sets''', because ''h''_''A'' reverses the direction of the morphisms, whereas ''h''^''A'' does not.