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)