# Bifunctor

A mapping , defined on the Cartesian product of two categories and with values in , which assigns to each pair of objects , some object , and to each pair of morphisms

the morphism

(1) |

The following conditions

(2) |

must also be met. In such a case one says that the functor is contravariant with respect to the first argument and covariant with respect to the second.

#### Comments

What is described above is a bifunctor contravariant in its first argument and covariant in its second. A bifunctor covariant in both arguments, the more fundamental notion ([a1]), has (1) and (2) replaced by

(1prm) |

(2prm) |

Similarly one can define bifunctors contravariant in both arguments and covariant in the first and contravariant in the second argument.

#### References

[a1] | B. Mitchell, "Theory of categories" , Acad. Press (1965) |

**How to Cite This Entry:**

Bifunctor.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Bifunctor&oldid=17051