The notion of a "category of Being" that Lawvere discusses there is the notion that more recently he has been calling a _category of cohesion_ . I'll try to illuminate a bit what's going on .

I'll restrict to the case that the category is a topos and say _cohesive topos_ for short. This is a topos that satisfies a small collection of simple but powerful axioms that are supposed to ensure that its objects may consistently be thought of as _geometric spaces_ built out of points that are equipped with "cohesive" structure (for instance topological structure, or smooth structure, etc.). So the idea is to axiomatize big toposes in which geometry may take place. 

Further details and references can be found here:

http://nlab.mathforge.org/nlab/show/cohesive+topos .

Let's walk through the article:

One axiom on a cohesive topos $\mathcal{E}$ is that the global section geometric morphism $\Gamma : \mathcal{E} \to \mathcal{S}$  to the given base topos $\mathcal{S}$ has a further left adjoint $\Pi_0 := \Gamma_! : \mathcal{E} \to \mathcal{S}$ to its inverse image $\Gamma^{\ast}$, which I'll write $\mathrm{Disc} := \Gamma^{\ast}$, for reasons discussed below. This extra left adjoint has the interpretation that it sends any object $X$ to the set $\Pi_0(X)$ "of connected components". What Lawvere calls a connected object in the article (p. 4) is hence one that is sent by $\Pi_0$ to the terminal object.

Another axiom is that $\Pi_0$ preserves finite products. This implies by the above that the collection of connected objects is closed under finite products. This appears on page 6. What he mentions there with reference to Hurewicz is that given a topos with such $\Pi_0$, it becomes canonically enriched over the base topos in a second way, a geometric way.

I believe that this, like various other aspects of cohesive toposes, lives up to its full relevance as we make the evident step to cohesive $\infty$-toposes. More details on this are here

http://nlab.mathforge.org/nlab/show/cohesive+(infinity,1)-topos

(But notice that this, while inspired by Lawvere, is not due to him.) 

In this more encompassing context the extra left adjoint $\Pi_0$ becomes $\Pi_\infty$ which I just write $\Pi$: it sends, one can show, any object to its geometric fundamental $\infty$-groupoid, for a notion of geometric paths intrinsic to the $\infty$-topos. The fact that this preserves finite products then says that there is a notion of _concordance_ of principal $\infty$-bundles in the $\infty$-topos.

The next axiom on a cohesive topos says that there is also a further right adjoint $\mathrm{coDisc} := \Gamma^! : \mathcal{S} \to \mathcal{E}$ to a total adjoint quadruple

$$
  (\Pi_0 \dashv \mathrm{Disc} \dashv \Gamma \dashv \mathrm{coDisc}) :=
  (\Gamma_! \dashv \Gamma^* \dashv \Gamma_* \dashv \Gamma^!) : \mathcal{E} \to \mathcal{S}
$$

such that both $\mathrm{Disc}$ as well as $\mathrm{coDisc}$ are full and faithful.

This is what Lawvere is talking about from the bottom of p. 12 on. The _downward functor_ that he mentions is $\Gamma : \mathcal{E} \to \mathcal{S}$. This has the interpretation of sending a cohesive space to its underlying set of points, as seen by the base topos $\mathcal{S}$. The left and right adjoint inclusions to this are $\mathrm{Disc}$ and $\mathrm{coDisc}$. These have the interpretation of sending a set of points to the corresponding space equipped with either _discrete cohesion_ or _codiscrete (indiscrete) cohesion_ . For instance in the case that cohesive structure is topological structure, this will be the discrete topology and the indiscrete topology, respectively, on a given set. Being full and faithful, $\mathrm{Disc}$ and $\mathrm{coDisc}$ hence make $\mathcal{S}$ a subcategory of $\mathcal{E}$ in two ways (p. 7), though only the image of $\mathrm{coDisc}$ will also be a subtopos, as he mentions on page 7. 

(This has, by the way, an important implication that Lawvere does not seem to mention: it implies that we are entitled to the corresponding quasi-topos induced by the sub-topos. That, one can show, may be identified with the collection of _concrete_ cohesive spaces. In the case of the cohesive topos for differential geometry, the concrete objects in this sense are precisely the _diffeological spaces_ .  )

He calls the subtopos given by the image of $\mathrm{coDisc} : \mathcal{S} \to \mathcal{E}$ that of "pure Becoming" further down on p. 7, whereas the subcategory of discrete objects he calls that of "non Becoming". The way I understand this terminology (which may not be quite what he means) is this:

whereas any old $\infty$-topos is a collection of _spaces_ , a cohesive $\infty$-topos comes with the extra adjoint $\Pi$ which I said has the interpretation of sending any space to its path $\infty$-groupoid. Therefore there is an intrinsic notion of _geometric paths_ in any cohesive $\infty$-topos. This allows notably to define parallel transport along paths and higher paths, hence a kind of _dynamics_ . In fact there is differential cohomology in every cohesive $\infty$-topos. 

Onwards. Notice next that every adjoint triple induces an adjoint pair of monads. In the present situation we get

$$
  (\mathrm{Disc} \;\Gamma \dashv \mathrm{coDisc}\; \Gamma) : \mathcal{E} \to \mathcal{E}
$$

This is what Lawvere calls the _skeleton_ and the _coskeleton_ on p. 7. In the $\infty$-topos context the left adjoint $\mathbf{\flat} := \mathrm{Disc} \; \Gamma$ has the interpretation of sending any object $A$ to the coefficient for cohomology of local systems with coefficients in $A$.

The paragraph wrapping from page 7 to 8 comments on the possibility that the base topos $\mathcal{S}$ is not just that of sets, but something richer. An example of this that I am kind of fond of is that of _super cohesion_ (in the sense of superalgebra and supergeometry): the topos of _smooth_ super-geometry is cohesive over the base topos of bare super-sets.

What follows on page 9 are thoughts of which I am not aware that Lawvere has later formalized them further. But then on the bottom of p. 9 he gets to the axiomatic identification of infinitesimal or formal spaces in the cohesive topos. In his most recent article on this what he says here on p. 9 is formalized as follows: he says an object $X \in \mathcal{E}$ is infinitesimal if the canonical morphism $\Gamma X \to \Pi_0 X$ is an isomorphism. To see what this means, suppose that $\Pi_0 X = *$, hence that $X$ is connected. Then the isomorphism condition means that $X$ has exactly one global point. But $X$ may be bigger: it may be a formal neighbourhood of that point, for instance it may be $\mathrm{Spec} \;k[x]/(x^2)$. A general $X$ for which $\Gamma X \to \Pi_0 X$ is an iso is hence a disjoint union of formal neighbourhoods of points.

Again, the meaning of this becomes more pronounced in the context of cohesive $\infty$-toposes: there objects $X$ for which $\Gamma X \simeq * \simeq \Pi X$ have the interpretation of being _formal $\infty$-groupoids_ , for instance formally exponentiated $L_\infty$-algebras. And so there is $\infty$-Lie theory canonically in every cohesive $\infty$-topos.

I'll stop here. I have more discussion of all this at:

http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+a+cohesive+topos