A category $\mathcal{C}$ with finite products is said to be Cartesian closed if each of the following functors has a right adjoint

1. ${0}:\mathcal{C}\to {1}$ where ${1}$ is the trivial category with one object $0$ and ${0}(A)=0$
2. the diagonal functor $\delta: \mathcal{C}\to \mathcal{C}\times\mathcal{C}$ where $\delta(A)=(A,A)$ and
3. for any object $B$ the functor $(-\times B):\mathcal{C}\to\mathcal{C}$ where $(-\times B)(A)=A\times B$ the product of $A$ and $B$

1. any functor ${1}\to\mathcal{C}$ where $0$ is mapped to an object $T$ in $\mathcal{C}$ $T$ is necessarily a terminal object of $\mathcal{C}$
2. the product (bifunctor) $(-\times -): \mathcal{C} \times \mathcal{C}\to \mathcal{C}$ given by $(-\times -)(A,B)\mapsto A\times B$ the product of $A$ and $B$
3. for any object $B$ the exponential functor $(-^B):\mathcal{C}\to\mathcal{C}$ given by $(-^B)(A)=A^B$ the exponential object from $B$ to $A$

In other words, a Cartesian closed category $\mathcal{C}$ is a category with finite products, has a terminal objects, and has exponentials. It can be shown that a Cartesian closed category is the same as a finitely complete category having exponentials.

Examples of Cartesian closed categories are the category of sets Set ( terminal object: any singleton; product: any Cartesian product of a finite number of sets; exponential object: the set of functions from one set to another) the category of small categories Cat (terminal object: any trivial category; product object: any finite product of categores; exponential object: any functor category), and every elementary topos.

Bibliography

1

S. Mac Lane, Categories for the Working Mathematician, Springer, New York (1971).