I read on [this n-lab page](https://ncatlab.org/nlab/show/full+and+faithful+functor) that a fully faithful functor $F: C\to D$ reflects all limits and colimits by the universal property.


On the other hand, I think a fully faithful functor does not always preserve limits and colimits since a "testing object" in $D$ is not necessarily in the image of $F$.


Are there any easy counter examples?



Does a fully faithful functor always preserve limits and colimits?
The forgetful functor from abelian groups to groups is fully faithful, and does not preserve coproducts. For example, in abelian groups, $\mathbb Z\coprod \mathbb Z=\mathbb Z\times \mathbb Z$, but in groups $\mathbb Z\coprod \mathbb Z$ is the free group on two generators.


