Bifunctor

A mapping $ T: \mathfrak A \times \mathfrak B \rightarrow \mathfrak C $, defined on the Cartesian product of two categories $ \mathfrak A $ and $ \mathfrak B $ with values in $ \mathfrak C $, which assigns to each pair of objects $ A \in \mathfrak A $, $ B \in \mathfrak B $ some object $ C \in \mathfrak C $, and to each pair of morphisms

$$ \alpha : A \rightarrow A ^ \prime ,\ \ \beta : B \rightarrow B ^ \prime $$

the morphism

\begin{equation} \label{eq1} T( \alpha , \beta ) : \ T(A ^ \prime , B) \rightarrow \ T(A, B ^ \prime ). \end{equation}

The following conditions

\begin{equation} \label{eq2} T(1 _ {A} , 1 _ {B} ) = 1 _ {T (A, B) } , \end{equation}

$$ T ( \alpha ^ \prime \circ \alpha , \beta ^ \prime \circ \beta ) = T ( \alpha , \beta ^ \prime ) \circ T ( \alpha ^ \prime , \beta ), $$

must also be met. In such a case, one says that the functor $T$ 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 \eqref{eq1} and \eqref{eq2} replaced by

\begin{equation} \label{eq1bis} \tag{1'} T ( \alpha , \beta ): \ T (A, B) \rightarrow \ T (A ^ \prime , B ^ \prime ), \end{equation}

\begin{equation} \label{eq2bis}\tag{2'} T ( \alpha ^ \prime \alpha , \beta ^ \prime \beta ) = T ( \alpha ^ \prime , \beta ^ \prime ) \circ T ( \alpha , \beta ). \end{equation}

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)