The notion of monodromy groupoid is used by  J. Pradines  togeneralise the standard construction of a simply connected Lie groupfrom a Lie algebra to a corresponding construction of a Lie groupoidfrom a Lie algebroid. 
Let $G$ be  a  topological groupoid  such that the fibres of initial point map of the groupoid are path connectedand have universal covers. Let  $\Mon( G)$ be  the disjoint union ofthe universal covers of the fibres at the base pointsidentities of the groupoid $G$.  Then there is a groupoid structureon $\Mon( G)$ defined by the concatenation composition of the pathsin the fibres.   The topological groupoid structures of $\Mon( G)$  are studied underin \cite{Br-Mu2}.  The groupoid  $\Mon(G)$ is called  {\em monodromy groupoid}  of $G$.
A {\em group-groupoid} is a group object in the category of groupoids; equivalently, itis an internal category and hence an internal groupoid  in the category of groups \cite{Por}.  An alternative name, quite generally used,  is  ``$2$-group''.  Recently the notion of monodromy for  topological  group-groupoids was  introduced and investigated in  \cite{Mu-Be-Tu-Na}.
In this paper,  the internal groupoid structure of monodromy groupoid for topological internal groupoids is developed.



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