@AlexNelson Ordinal exponentiation is about something else altogether.

Scott's models are directed-complete partial orders (DCPOs), which form a cartesian closed category, and the exponential $A^B$ of two DCPOs does not consist of all functions from $B$ to $A$, but only those which preserve directed joins. Therefore it is not correct to say that $|A^B| = |A|^{|B|}$ (the cardinal exponential on the right counts all functions). I may come back to say more about the question, but there is nothing shocking going on.

Regarding the last paragraph, it's true that the forgetful functor to sets is not monadic, because it does not reflect isomorphisms. A simple example is the evident poset bijection from $a \leq c \geq b$ to $0 \leq 1 \leq 2$, which preserves directed joins.

Usually when someone asks on MO whether a functor has an adjoint and the answer is not obvious, I reflexively go for the contrapositive of (a).

@Ceecee No, inmates generally do not have access to the internet. But someone can print out and mail articles from arxiv in any chosen subject area, so the suggestion is good.

@MonroeEskew As Andres and Paul are suggesting, it's hard to rule out that mathematics might have some bearing on the matter, and much in logic that was once considered as "belonging to" philosophy has been mathematicized. The question is not a beginner question and merits consideration.

@PaulTaylor Yes, but it seems OP is working in a Ab-enriched context. It's additivity (existence of direct sums) plus splitting of idempotents which is operative here: the absolute colimit completion.

I'm not qualified to answer, but could you give the publication data of the Suslin paper? It looks like an interesting question (I am supposing that "unfolding" means something like "beta reduction" in the informal sense ncatlab.org/nlab/show/beta-reduction#informal_usage). I hope the LaTeX didn't introduce any errors.

See also Sketches of an Elephant, D.4.7.12.

Richard Borcherds is "only" a Fields Medalist (1998).