What is the adjunct of the evaluation morphism in a closed monodical category? According to the nlab article about evaluation map if $X, Y \in C$, a closed monodical category then the adjunct to evaluation morphism $[X, Y]\otimes X \rightarrow Y$ is the identity morphism $[X, Y] \rightarrow [X, Y]$.

Is that correct and if yes why?
Because it seems to me that the adjunct of such has to be $X \rightarrow [Y, (X \otimes Y)]$
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 $GA = [X,A]$.