Though this might not be what you are expecting, I will explain you "oidification" or horizontal categorification as I understood (Experts are fell free to add or edit as necessary). This is the process that generalizes a "certain type of category with a single object" to "such type of categories with multiple objects". This is done mostly via "enriching" the initial category $\mathcal{C}$ over another monoidal category $\mathcal{K},$ which roughly says homsets (set of arrows two objects) of $\mathcal{C}$ are replaced by objects of $\mathcal{K}.$
Examples include
| X | X-oid | Enrichment |
|---|---|---|
| monoid | Category | categories enriched over Set |
| Category | 2-Category | categories enriched over Cat |
| Group | Groupoid | |
| Ring | Ringoid | category enriched in tensor category Ab |
| Quantale | Quantaloid | category enriched in suplattices |
| Algebr | Algebroid | category enriched in Vect or RMod |
| C*-algebra | C*-category | *-category enriched in Ban |
You can find more details ringoid, and algebroid here. But as far as I know Hopf algebroids and Lie algebroids does not fit into this general definition of algebroids, but still multi-object generalizations of their counterparts.
