Exponentiable object

An object $X$ in a category with finite products is exponentiable when $-\times X$ has a right adjoint $[X,-]$. The terminal object is exponentiable, and exponentiable objects are closed under binary products because right adjoints compose:

In a complete category that is a locally small category with a coseparator $S$ and all monomorphisms regular, an object $A$ is an exponentiable object exactly when $\mathcal{C}(-\times A,S)$ is a representable functor. A representing object $E$ gives representations $E^I$ for maps into powers $S^I$. An equalizer presentation by powers of a coseparator then constructs a representation for maps into every target. The Yoneda lemma makes these representations functorial and supplies the right adjoint.

If $A$ and $B$ are exponentiable objects, thenhas the composite of their exponential functors as a right adjoint. The terminal object supplies the empty product, so exponentiable objects are closed under finite products.

If the terminal object is also initial and $E$ is exponentiable, the left adjoint $-\times E$ preserves the initial object. But $0\times E\cong E$ because $0$ is terminal, while preservation of initiality gives $0\times E\cong 0$. Hence $E\cong 0$.

A $T_0$ space $E$ is exponentiable in ${\mathbf{Top}}_0$ if and only if the functor ${\mathbf{Top}}_0(-\times E,S)$ to sets is representable, where $S$ is the Sierpiński space. Every $T_0$ space is an equalizer of maps between powers of $S$, so a representing object for maps into $S$ constructs exponentials for every target by products and equalizers.