Exponential object

Let $\mathcal{C}$ be a category with binary products (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations). For objects $B,C$, an exponential object of $C$ by $B$ is an object $C^B$ together with an evaluation morphism

$\mathrm{ev}:C^B\times B\to C$

such that for every object $A$ and every morphism $f:A\times B\to C$ there is a unique morphism

$\lambda f:A\to C^B$

with

$\mathrm{ev}\circ (\lambda f\times 1_B)=f\mathrm{.}$

If exponential objects $D^B$ are chosen for every object $D$ and these choices are assembled functorially, their universal properties say equivalently that $D\mapsto D^B$ is right adjoint to $-\times B$; in particular its value at $C$ is $C^B$. The existence of the single exponential $C^B$ alone does not assert that this right adjoint exists on every object.