Fix a universe $\mathcal{U}$. Call a category $\mathcal{U}$-complete if every diagram indexed by a $\mathcal{U}$-small category has a limit, and a functor $\mathcal{U}$-continuous if it preserves $\mathcal{U}$-small limits. Usually, when one fixes a universe, one calls this simply complete and continuous.


Now assume we are given $\mathcal{U}$-complete categories $C,D$ and a $\mathcal{U}$-continuous functor $F : C \to D$. Does then $F$ also preserve limits (which exist in $C$) which are not necessarily $\mathcal{U}$-small?



Colimits in a bigger universe
No. I'll answer for the case of colimits as follows: Consider for example the category $O$ of ordinals (as a poset), and adjoin a terminal object $T$, making a larger category $C$. Then this terminal object is a (large) colimit over the diagram $O\to C$. However, a cocontinuous functor $C\to D$ can send $T$ anywhere that admits a cone under the diagram $O\to D$, not necessarily its colimit (if the colimit even exists).


