  Let $\Bbbk$ be a commutative ring and $I$ a category. As a generalization of
a $\Bbbk$-category with a (pseudo) action of a group we consider a family of
$\Bbbk$-categories with a (pseudo, lax, or oplax) action of $I$, namely an
oplax functor from $I$ to the 2-category of small $\Bbbk$-categories. We
investigate derived equivalences of those oplax functors, and establish a
Morita type theorem for them. This gives a base of investigations of derived
equivalences of Grothendieck constructions of those oplax functors.
