Exponential object

Remarks

The morphism $\mathrm{ev}$ is the evaluation, and $\lambda g$ is the transpose or curried form of $g$. An exponential object is unique up to a unique isomorphism compatible with $\mathrm{ev}$. If $Z^Y$ exists for all $Z$, then $Z \mapsto Z^Y$ is a functor right adjoint to $- \times Y$.

Example: in the category of sets, let $Z^Y$ be the set of maps $Y \to Z$ and $\mathrm{ev}(h, y) = h(y)$. For $g \colon X \times Y \to Z$, the map $\lambda g$ sends $x$ to $y \mapsto g(x, y)$. It is the only map with $\lambda g(x)(y) = g(x, y)$ for all $x, y$.

Non-example: in the category of topological spaces, not every exponential exists. For the space $\Q$ of rational numbers, the functor $- \times \Q$ has no right adjoint, so some $Z^{\Q}$ do not exist. For $Y$ locally compact Hausdorff, $Z^Y$ exists and is the space of maps with the compact-open topology.