$ \newcommand{\Span}{\mathbf{Span}} \newcommand{\cE}{\mathcal E} $Given a category $\cE$ with plenty of limits, let $\Span(\cE)$ denote the bicategory of spans in $\cE$. It is known that monads in $\Span(\cE)$ are the same thing as internal categories in $\cE$ (this is not quite true at the level of natural transformations, but this can be fixed, see Street's "Formal Theory of Monads II"). A monad $t\circlearrowright C$ in a reasonable 2-category has a corresponding Eilenberg-Moore object $C^t$ of algebras; for example, this can be given by a weighted limit formula.

If I view an internal category $C_\bullet$ in $\cE$ as a monad in $\Span(\cE)$, what's the corresponding Eilenberg-Moore object? I couldn't find this in the literature, and seemed to get nonsense when I tried working it out myself.