Cartesian closed category

Remarks

Clauses 1 and 2 say that $\mathcal{C}$ has finite products. Clause 3 then says that each functor $- \times Y$ has a right adjoint $(-)^Y$. So there are bijections $\Hom(X \times Y, Z) \cong \Hom(X, Z^Y)$, natural in $X$ and $Z$.

Example: the category of sets is Cartesian closed. A one-point set is terminal, $X \times Y$ is the Cartesian product, and $Z^Y$ is the set of maps $Y \to Z$ with evaluation.

Non-example: the category of abelian groups is not Cartesian closed, although clauses 1 and 2 hold. In a Cartesian closed category, $- \times Y$ is a left adjoint, so it preserves the initial object $0$, and $0 \times Y \cong 0$. In abelian groups $0 \times \Z \cong \Z$, which is not $0$, so clause 3 fails.