The general notion you're looking for is a representable functor. For example:

$${\mathcal E}(X\times{-},Y) \sim {\mathcal E}({-},Y^X)$$

$${\textsf{Sub}}({-}) \sim {\mathcal{E}}({-},\Omega)$$

The thing on the left is a general contravariant functor from the category to $\mathbf{Set}$.

The thing on the right is of the form ${\mathcal E}({-},R)$, where $R$ is the "representing object" for the functor.

The general notion you're looking for is a representable functor. For example:

$${\mathcal E}(X\times{-},Y) \sim {\mathcal E}({-},Y^X)$$

$${\textsf{Sub}}({-}) \sim {\mathcal{E}}({-},\Omega)$$

The thing on the left is a general contravariant functor from the category to $\mathbf{Set}$.

The thing on the right is of the form ${\mathcal E}({-},R)$, where $R$ is the "representing object" for the functor.

The definition of a préfaisceau représentable is given in SGA4 Exposé I, remark 1.4.2. Presumably the idea is then used, maybe extensively, in SGA4 and beyond, but I will leave others to search for it.

This is also one of the many equivalent ways of expressing an adjunction.

Of course, whether or not the (topos or other) category has a representing object for a particular functor is very much up for debate in the topic in question.