Equivalences of derived categories of algebras and Lie algebras realized by Ringel-Hall algebras of algebras

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Let k be a commutative ring, I a small category and denote by k-Cat the 2-category of small k-categories. For 2-categories B, C, denote by Colax(B,C) the 2-category consisting of the colax functors from B to C and the lax transformations between them. We proved that each pseudofuncor C→D of 2-categories induces a pseudofunctor Colax(B,C)→Colax(B,D) for all 2-categories B, and defined the "module category", the "derived category", and derived equivalences of colax functors I→k-Cat by using it. We proved that if two such colax functors are derived equivalent, then so are their Grothendieck constructions, by which we gave a way to glue derived equivalences together.