Universal categories

The category of models of any theory $T$ in any first-order language $L$ has the surprising property that any small category that is elementarily equivalent with it, already embeds in it. The proof uses an abstract argument via ultrapowers, leaving one wonder which concrete categorical axioms, depending on $T$ and $L$, are responsible for this embedding result. We also propose a first-order logic for which equivalent categories are always elementarily equivalent.