As Todd has already written an answer for me, maybe I can claim it as an Answer:

Exercise 1.1 in my book *Practical Foundations of Mathematics*
(CUP 1999) reads,

When Bo Peep got too many sheep to see where each one was throughout
the day, she found a stick or a pebble for each individual sheep and
moved them from a pile outside the pen to another inside, or vice
versa, as the corresponding sheep went in or out.
Then one evening there was a storm, and the sheep came home too
quickly for her to find the proper objects, so for each sheep coming
in she just moved ANY one object. She moved all of the objects, but
she was still worried about the wolf. By the next morning she had
satisfied herself that the less careful method of reckoning was
sufficient. Explain her reasoning **without the aid of numbers**.

My reason for putting it in the book was to try to get some
anthropologist to say when and what the original "proof" was, ie the
cognitive basis of the long-universal belief that this is valid, which
provides the justification of counting with numbers.

What I am trying to imagine is how one of our distant ancestors
with the cognitive abilities but not the education of a mathematician
might approach this. Of course they would not have formulated
Peano Induction or Euclidean Infinite Descent. They would have
an argument (that we would more or less accept as rigorous) that
the result is true for three sheep, then four and five. After
that they would use Induction in the naive epistemological sense
to convince themselves that it is true for arbitrarily large sets.

It seems plausible that anyone who is challenged to come up with
a proof would give the following (albeit non-constructive) proof.

Suppose that some sheep $s_0$ is missing in the evening.
Then the pebble $p_0$ that served as its "name" in the morning was
used for some other sheep $s_1$ in the evening.
But then $s_1$ must have been named by a different pebble $p_1$
in the morning, which named yet another sheep $s_2$ in the evening.
And so on.
All of the sheep $s_0$, $s_1$, $s_2$, ... are different individuals,
Likewise all of the pebbles $p_0$, $p_1$, $p_2$, ...
But, essentially as Euclid says in Book VII, Proposition 31,
this is impossible for a set of sheep.

By chance, this issue came up recently following an internal seminar by
Martin Escardo in Birmingham (where I am now an Honorary Research
Fellow).
He was developing the foundations of arithmetic (in the setting
of Homotopy Type Theory, though this was not essential)
in such a way that $3\times 5=5\times 3$ could be seen in
a primary-school fashion as transposing a rectangle.

He based this on a function $F:{\mathbb{N}}\to{\mathsf{Set}}$
with $F0=\emptyset$ and $F(\mathsf{succ} n)=F(n)\coprod{\mathbf{1}}$.
In his treatment the most difficult Proposition is
$$ F(n)\cong F(m) \Longrightarrow n=m, $$
which he deduced from the Lemma
$$ X\coprod{\mathbf{1}}\cong Y\coprod{\mathbf{1}}
\Longrightarrow X \cong Y. $$

This Lemma holds in any **lextensive category**,
*i.e.* one with finite limits and stable disjoint coproducts.
The Proposition follows using Peano induction, since then
$$ F(n+1)\cong F(m+1) \Longrightarrow F(n)\cong F(m)
\Longrightarrow n=m \Longrightarrow n+1=m+1. $$

I think it is reasonable to suppose that Bo Peep could formulate
this Lemma, but I feel it is more plausible that she
would use the "infinite descent" argument than the Proposition.

I am not sure whether this answers the original question
about justifying "the use of categories, or isomorphisms or equivalences",
although maybe Martin's treatment of arithmetic does so.