What is the adjunct of the evaluation morphism in a closed monodical category?
Adjuncts are morphisms related by the natural isomorphism:

$$Hom(FA, B) \cong Hom(A, GB)$$

defining the adjunction. The adjuncts of (co)units are identities:

$$
(ε_A : FGA → A)  \longleftrightarrow (1_{GA} : GA → GA) \\
(η_A : A → GFA) \longleftrightarrow (1_{FA} : FA → FA)
$$

This correspondence is basically what connects the $Hom$ definition of adjunctions with the (co)unit definition.

In this case, $FA = A \otimes X$ and $GA = [X,A]$.