According to the nlab article on the evaluation map if $ X, Y in C $, a closed monodic category and then the complement to the evaluation morphism $ (X, Y) otimes X rightarrow Y $ is the morphism of identity $ (X, Y) right arrow (X, Y) $.

Is it correct and if so, why?

Because it seems to me that the complement of such has to be $ X rightarrow (Y, (X otimes Y)) $