What is the internal hom functor in the context of an internally projective object?

Question

I am trying to understand the definition of an internally projective object from nLab.

It says that an object $E$ of a topos $\mathcal{T}$ is called internally projective if the internal hom functor $(−)^E:\mathcal{T} \to \mathcal{T}$ preserves epimorphisms.

My confusion is with the definition of the internal hom functor. When I click on its definition, then the internal hom functor is denoted as $[-,-]:\mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}$ which has a different type from $\mathcal{T} \to \mathcal{T}$. Or could I regard a topos as both of type $\mathcal{C}$ and of type $\mathcal{C}^{op} \times \mathcal{C}$?

So what is the definition of the internal hom functor in the context of an internally projective object?

Answer 1

Using that notation, $(-)^E$ is $[E, -]$. The internal hom is what you get from cartesian closure.