An internal bigroupoid (in $\mathrm{Top}$, for argument’s sake, but any finitely complete category will do, or at least one with some pullbacks, like $\mathrm{Diff}$) consists of the following data:
A space ${B}_{0}$,
An internal groupoid ${\underline{B}}_{1}=({B}_{2}\rightrightarrows {B}_{1})$ equipped with a functor
A (horizontal) composition functor
over ${B}_{0}\times {B}_{0}$
A unit functor
over ${B}_{0}\times {B}_{0}$
A (horizontal) inverse functor
covering the swap map from ${B}_{0}\times {B}_{0}$ to itself.
Together with natural transformations… (see for the time being Definition 5.21 in my thesis - I need to grok how to do diagrams here)