TITLE: How different can equivalent categories be?
QUESTION [28 upvotes]: Some pretty big results in mathematics can be expressed as equivalence of categories (or rather, as adjunctions). The usual definition $FG\cong 1,GF\cong 1$ is easily justifiable in category theory since equality is too strict for objects of a category (functor categories in this case).
While looking at Borceux (Vol I, paragraph after prop 3.4.5), I saw for instance that equivalent categories may have non-isomorphic sets of objects (the functor $N:\mathbf 2\rightarrow \mathbf 1$). What are some explicit examples of interesting equivalent categories with non-isomorphic classes of objects?
More generally:

So exactly how different can equivalent categories be?

This MSE question seems to be very much related, both to my question and to the remark in Borceux about equivalent categories not being interchangeable "from outside". However, the level of discussion there is too high for me to really feel I understand exactly what's going on...
Intuition, instructive examples, and formal results are all welcome.

REPLY [10 votes]: (Warning: I'm writing this from memory, and don't have the reference handy)
In Categories, Allegories, Freyd and Scedrov prove a theorem

A property of categories is preserved by equivalence if and only if it can be described on a blackboard

The key feature of a blackboard, here, is that if you draw two distinct objects, they remain distinct throughout a description; you can't insist they are equal. So we might restate the theorem as

A property of categories is preserved by equivalence if and only if it does not involve equations of objects

To prove this, Freyd and Scedrov define a formal diagrammatic language whose semantics involve the existence of functors from diagram categories. For example, limits from a category $J$ are described as follows:

Define a category $J_1$ by adjoining to $J$ a new object $L$, arrows $\hom(L,L) = \hom(L, j) = \{ * \}$
Define a category $J_2$ by adjoining to $J_1$ a new object $M$ and arrows $\hom(M,M) = \hom(M,j) = \{ * \}$ and $\hom(L,M) = \hom(M,L) = \varnothing$
Define a category $J_3$ by adjoining to $J_2$ a new arrow $p : M \to L$

Then, a functor $F : J \to \mathcal{C}$ has a limit if and only if:

There exists a a functor $F_1 : J_1 \to \mathcal{C}$ extending $F$ such that
For every functor $F_2 : J_2 \to \mathcal{C}$ extending $F_1$
There exists a unique functor $F_3 : J_3 \to \mathcal{C}$ extending $F_2$.

In the formal version of the claimed theorem, the relevant property is that the functors between the diagram categories must be injective on objects.
An example of a property not preserved by equivalence is as follows:

Let $J_1$ be the category of two objects and an isomorphism between them
Let $J_2$ be the category consisting of a single object and $\hom(*, *) = \mathbb{Z}$

And map $J_1 \to J_2$ by sending the two isomorphisms to $\pm 1$. Then, a category $\mathcal{C}$ is skeletal if and only if every functor $J_1 \to \mathcal{C}$ can be extended to a functor $J_2 \to C$.

With the benefit of hindsight, I think the key idea underpinning this is the canonical model structure. In particular, consider the square
$$ \begin{matrix} J &\xrightarrow{f}& \mathcal{C}
\\ i\downarrow\ \  & & \ \ \downarrow e
\\ J' &\xrightarrow{f'} & \mathcal{C}' \end{matrix} $$
A lift, here, is a diagonal arrow $\ell : J' \to \mathcal{C}$ making the diagram commute.
Then $i$ is injective on objects if and only if, for every such diagram where $e$ is a surjective equivalence $e$, a lift exists.
Similarly, $e$ is a surjective equivalence if and only if, for every such diagram where $i$ is injective on objects, a lift exists.
The connection to the logic described above are, if $i$ is injective on objects and $e$ a surjective equivalence, then

A functor $f : J \to \mathcal{C}$ can be extended to $\ell : J' \to \mathcal{C}$ if and only if $ef : J \to \mathcal{C}'$ can be extended to $f' : J' \to \mathcal{C}'$. Furthermore, in both cases the lifts can be chosen so that they are related via $e$.

This extends to the general case, since any equivalence can be 'lifted' to a span of surjective ones; for any equivalence $e:\mathcal{C} \to \mathcal{C}'$, there is a category $\mathcal{C}''$ and surjective equivalences $e_1 : \mathcal{C}'' \to \mathcal{C}$ and $e_2 : \mathcal{C}'' \to \mathcal{C}'$ such that $e e_1 = e_2$.