I am searching for a category $\mathcal C$ defined by a species of structures with morphisms $\Sigma$ (here I mean what is called 'espèce de structure' in Bourbaki Set Theory, chapter IV; put simply: the objects of $\mathcal C$ are sets endowed with some structure and morphisms are set-theoretic maps that respect this structure) such that the following holds:
- direct limits (taken over directed sets) exist in $\mathcal C$;
- the underlying set of a direct limit in $\mathcal C$ is not in general the direct limit of the underlying sets.
Please ask me if something is not clear. Thanks in advance.