Question about the univalence axiom versus skeleta. Here, Dan Licata writes:

[Univalence] can be used to build algebraic structures in such a
  way that isomorphic structures are equal (e.g. equality of groups is
  group isomorphism).

He writes something similar here:

A consequence of the univalence axiom is that isomorphic types are
  equivalent (propositionally equal).

Now nLab writes:

A category is skeletal if objects that are isomorphic are necessarily
  equal; so this is a notion irredeemably violating the principle of
  equivalence of category theory.

Is there not a contradiction here? How can we claim that:


*

*skeletons violate the principle of equivalence

*univalence upholds it


when apparently, univalence implies that every category is skeletal?
 A: Well, Homotopy Type Theory introduces some extraordinary equality, which approach requires special attention, and so the n-Lab article is a little bit out of this context, as it uses categories over some classical set theory, while HoTT defines its own 'precategories' and 'categories'.
For a motivating example, when we arrange four small balls in $2\times 2$ matrix, we give an interpretation of the multiplication $2\times 2$ of numbers, but as a number this arrangement equals to $4$:
$$2\times2\ =\ 4 $$
Similarly, we might want that e.g. the Klein group $V_4$ and the product group $\Bbb Z_2\times\Bbb Z_2$ would be equal, as groups (independently of exactly which four elements build them and how).
$$\Bbb Z_2\times\Bbb Z_2\ =\ V_4$$
As you have found, categories --as defined in HoTT-- are indeed necessarily skeletal in that setting.


*

*That 'Skeletons violate the principle of equivalence' refers to the fact that the property 'skeletal' is not invariant under category equivalence.

*As every category (in the sense and context of HoTT[!]) is skeletal, category equivalence here seems to coincide to category isomorphism, hence -by Univalence- to equality. 
In HoTT the notion 'skeletal' just gets meaningless.
But the whole HoTT theory is still under development, terms and definitions are not necessarily in their finalized form yet..
