Notice that by definition of inner hom, $(-)^\Delta$ always has a left adjoint. A right adjoint can only exist for very particular objects. Therefore the term amazing right adjoint

The $Y_\Delta$ defined this way is indeed a sheaf, due to the assumption that $(-)^\Delta$ preserves colimits. So this is indeed a right adjoint.

Related concepts

A topos$\mathcal{X}$ is a local topos (over Set) if its global section functor $\Gamma = Hom(\ast_{\mathcal{X}}, -)$ admits a right adjoint. This is hence an “external” version of the amazing right adjoint, exhibiting $\ast_{\mathcal{X}}$ as “atomic”.