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$ \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.

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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.

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Suppose that $A$ is a monad on the object $C$ in $\Span(\cE)$. So, we have an internal category in $\cE$, also denoted by $A$.

An EM-object of $A$ is just the universal $A$-algebra. Algebras between monads in $\Span(\cE)$ are internal profunctors, see https://ncatlab.org/nlab/show/internal+profunctor. While a (one-sided) $A$-algebra consists of an object $D$ of $\cE$ and an internal profunctor from $D$ considered as a discrete internal category (trivial monad in $\Span(\cE)$) to the category $A$. So, an $EM$-object of $A$ consists of an object $P$ of $\cE$ together with an internal profunctor from $P$ (considered as a discrete internal category) to $A$, which induces

$$\mathbf{Prof}(D, A) \cong \cE(D, P)$$

With the existence of "internal presheaf" object $\mathcal{P}A$

$$\mathbf{Prof}(D, A) \cong \cE(D, \mathcal{P}A)$$

and than we can take $P = \mathcal{P}A$. The idea of "internal presheaf" goes to the notions of cosmos and Yoneda structures, for which you can see https://link.springer.com/chapter/10.1007/BFb0063103 and http://www.academia.edu/30587479/Yoneda_structures_on_2-categories. In particular, for "internal presheaf" $\mathcal{P}A$ for an internal category $A$ see Examples (2) in the latter paper. I believe following this direction might lead to precise formulations.

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