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A diagram consisting of differential graded (dg for short) categories and dgfunctors is formulated as a colax functor $X$ from a small category $I$ to the2-category $\kdgCat$ of small dg categories, dg functors and dg naturaltransformations for a fixed commutative ring $\k$. If $I$ is a group regardedas a category with only one object $*$, then $X$ is nothing but a colax actionof the group $I$ on the dg category $X(*)$. In this sense, this $X$ can beregarded as a generalization of a dg category with a colax action of a group.We define a notion of standard derived equivalence between such colax functorsby generalizing the corresponding notion between dg categories with a groupaction. Our first main result gives some characterizations of this notionwithout an assumption of $\k$-flatness (or $\k$-projectivity) on $X$, one ofwhich is given in terms of generalized versions of a tilting object and aquasi-equivalence. On the other hand, for such a colax functor $X$, the dgcategories $X(i)$ with $i$ objects of $I$ can be glued together to have asingle dg category $\Gr(X)$, called the Grothendieck construction of $X$. % Oursecond main result insists that for such colax functors $X$ and $X'$, theGrothendieck construction $\Gr(X')$ is derived equivalent to $\Gr(X)$ if thereexists a standard derived equivalence from $X'$ to $X$. These resultsgeneralize the main results of \cite{Asa-a} and \cite{Asa-13} to the dg case,respectively. These are new even for dg categories with a group action. Inparticular, the second result gives a new tool to show the derived equivalencebetween the orbit categories of dg categories with a group action, which willbe illustrated in some examples.

Characterizations of standard derived equivalences of diagrams of dg categories and their gluings