Cahoone here emphasizes very different themes than we talked about on the episode, specifically the theistic themes (he characterizes "Spirit" as "pantheistic" or "panentheistic," both of which have been used to describe Spinoza; the former means everything is God, while the latter means everything is within God, but God can exceed creation as we're aware of it) as well as politics.
Hegel's Phenomenology, according to Cahoone, involves detailing the shapes or forms of Spirit (geist) as they evolve in human history. "God is evolving, and human beings are part of that evolution, by which God comes to full self-consciousness or self-recognition."
He characterizes dialectic: "the manner of transition or development." "We begin with something or other... which given the context in which it operations naturally generates opposition or conflict with something else. This conflict eventually leads to a resolution: a transformation of the opposed elements to form a new unity." Anything finite must undergo the dialectic, i.e. anything except God in his total realization.
From part two: Reason is what gets at the inner movement of things, i.e. doing dialectical analysis just like he's doing in the book. Understanding, which is just what natural science does, just records regularities. This is much like Schopehauer's distinction between the field of objects analyzable using the principle of sufficient reason in its various forms--the objects and forms of experience and concepts extracted from them--and analysis of nature in terms of Will, which Schopenhauer sees as necessary for understanding not only other people, but any other life and really any force at all.
Hegel's theory of truth, according to Cahoone (around 5 minutes into part 2), is not the correspondence of ideas to reality, but the coherence of ideas into the whole. Unlike for the pragmatists, this is not just a matter of the coherence of ideas in my head or with some other set of beliefs held in a community, but coherence with the whole of reality. Now, Schopenhauer, of course, would dismiss this mixing of beliefs with things that are not beliefs as nonsense; this is all in keeping with Hegel's desire (elaborated further in our Heidegger episode, as Heidegger took this theme and ran with it) to remove the hard distinction between ideas locked in peoples heads and the contents of a purely material world. I don't think Cahoone gives this enough time in this particular lecture to make the idea clear.