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 $\Gamma = Hom(1_{\mathcal{X}}, -)$ admits a right adjoint. This is hence an “external” version of the amazing right adjoint, exhibiting $1_{\mathcal{X}}$ as “atomic”.

References

The ubiquity of right adjoints to exponential functors in the context of synthetic differential geometry was first pointed out in Lawvere (1980). Lawvere (2004) suggests to augment lambda calculus with such fractional operators. Thorough discussion of the concept is in Yetter (1987) and Kock&Reyes (1999). Moerdijk&Reyes (1991) have a succinct overview in the context of SDG as does Lawvere (1997).

William Lawvere, Toward the Description in a Smooth Topos of the Dynamically Possible Motions and Deformations of a Continuous Body, Cah.Top.Géom.Diff.Cat. 21 no.4 (1980) pp.377-392. (pdf)