Bounded geometric morphisms, origin and motivation for the terminology

Question

Bounded geometric morphisms serve as a generalization of Grothendieck topoi; $T$ being a Grothendieck topos iff global section $T \rightarrow Set$ is bounded. I managed to only track this down to the Johnstone's, 1977 Topos theory, where it is defined, but the terminology "bound"/"bounded" isn't motivated. The definition of Bounded geometric morphism 4.43, is attributed to "W. Mitchell", but none of the 3 references by Mitchell (William Mitchell, the only Mitchell in the bibliography as a sole author) in the bibliography seem to use the terms "bound"/"bounded" or discuss a similar definition on a skim.

This results in 3 questions:

1. What's the motivation for calling these geometric morphisms bounded, calling part of the definition a "bound"?
2. What's the original text defining and hopefully motivating the terminology "bounded"?
3. Is there a precise connection with order theoretical bounded?

The 3 references by William Mitchell being:

1. "Categories of Boolean Topoi"
2. "On topoi as closed categories"
3. "Boolean topoi and the theory of sets"

Answer 1

Here is what Peter Johnstone wrote on the matter in response to me passing on the question:

I was certainly around at the time that terminology came into use, and I may have been the first person to use it in print (in my 1977 book). In his 1975 paper "Change of base for toposes with generators", Radu Diaconescu referred to a topos F "having generators" over a base E, but that seemed a bit cumbersome; the single word "bounded" had the right overtones, in that it referred to a single object of F (which I later took to calling simply a "bound") that controlled how complicated the objects could be relative to those of E. I don't know whether I actually suggested the term, but as far as I can recall everyone agreed that it was a sensible choice.

Incidentally, the Grothendieck school were aware of the existence of what we would call unbounded Set-toposes, only they didn't consider them to be toposes. (There is one described in SGA4 as a "faux topos".)

Answer 2

A bounded geometric morphism $f:\mathcal E\to \mathcal S$ is one that has a bound $B$, i.e. an object such that all of $\mathcal E$ is "below" $f^*(\mathcal S)$ relative to $B$, in the sense of being a subquotient of $f^* S\times B$ for some $S.$ In motivating examples, $B$ might be constructed as a coproduct over a small set of generating objects, as for instance in the Artin gluing of an accessible Set-functor, whose bound is explicitly constructed using a coproduct of sets smaller than the cardinal bound $\kappa.$ Similarly, the bound for a Grothendieck topos can always be taken as the coproduct of all $\kappa$-presentable objects for sufficiently large $\kappa.$ So you might imagine boundedness is intended to call up the intuition of a cardinal bound in this sense. I do not know whether there's an appropriate notion of indexed local presentability making this intuition more precise, nor whether this is historically accurate; I personally think that the properties of $B$ as given at the beginning of my answer have the feel of a bound without going any further.