The categories Set and Ens
In Categories for the working mathematician, Mac Lane assumes the existence of one Grothendieck universe $U$, and $\mathbf{Set}$ is the category of sets in $U$. This device ensures the existence of functor categories like $[\mathbf{Set}, \mathbf{Set}]$.
On the other hand, $\mathbf{Ens}$ is any full subcategory of the metacategory of all sets, with the restriction that the collection of objects in $\mathbf{Ens}$ is itself a set. Note that $\mathbf{Ens}$ may fail to have the properties expected of $\mathbf{Set}$, e.g. cocompleteness, cartesian closedness, etc.