$
\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.



Algebras in a bicategory of spans
In general, it doesn't exist. That is, $\mathbf{Span}(\mathcal{E})$ doesn't have all EM-objects. If we embed $\mathbf{Span}(\mathcal{E})$ into the bicategory $\mathbf{Prof}(\mathcal{E})$ of internal categories and profunctors, then the EM-object of an internal category $C$ regarded as a monad in $\mathbf{Prof}(\mathcal{E})$ coincides with its Kleisli object, and is $C$ itself, regarded instead as an object of $\mathbf{Prof}(\mathcal{E})$. This makes it easy to see that most EM-objects in $\mathbf{Span}(\mathcal{E})$ do not exist: if $C$ had an EM-object in $\mathbf{Span}(\mathcal{E})$ then $C$ would be Morita equivalent to a discrete internal category.


In fact, $\mathbf{Prof}(\mathcal{E})$ is the "free cocompletion" of $\mathbf{Span}(\mathcal{E})$ under Kleisli objects (and hence also the free completion under EM-objects, since both are self-dual) in a suitable sense. The Lack-Street paper FTM2 that you mention is one important ingredient in this (it shows how to get the 2-cells right), but in order to get the *1-cells* right one needs to look at a suitable kind of enriched cocompletion; see [arXiv:1301.3191](https://arxiv.org/abs/1301.3191).


