 \\ & = & {\operatorname{\bf R}}\pi_* {\operatorname{\bf L}}(j \times {\operatorname{Id}}_E)^* {\operatorname{\bf R}}{\curly Hom}({\mathbb{I}}^\bullet, {\mathbb{I}}^\bullet)[0] \\ & = & {\operatorname{\bf R}}\pi_* {\operatorname{\bf R}}{\curly Hom}( {\operatorname{\bf L}}(j \times {\operatorname{Id}}_E)^* {\mathbb{I}}^\bullet