Don't waste time, just write the paper.

• reflexive and coreflexive

Suppose $i: D \into C$ is a full subcategory, and the inclusion preserves colimit, then we have the right-adjoint of $i$. Denoted as $R_D$. Suppose $D$ is generated by one object $d$, $$ R_D = d \otimes_{End(d)} Hom(d, -) : C \to D $$ everything is derived.