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In my category theory course lecture notes the following theorem:

Theorem: Every presheaf (of sets) on a small category $mathcalC$ is a colimit of representables. More precisely, for each presheaf $F$ we have an isomorphism:
$$ F cong textcolim_(astRightarrow F)^op P $$
for $(ast Rightarrow F)$ the comma category and $P(x, y) := Hom(-, x)$.

is followed up by the following note and theorem:

Note: One says that the presheaf category is the ''free cocompletion'' of $mathcalC$. It is not true that the formation of presheaves is a left adjoint to some forgetful functor due to size issues. [Italics mine]. We do however have:

Theorem: Let $mathcalC$ be a small category and $mathcalD$ a category with all small colimits. Then there is an equivalence of categories:
$$Fun^colim(Preshvs(mathcalC), mathcalD) cong Fun(mathcalC, mathcalD)$$
where the catgeory on the left is the full subcategory of functors preserving small colimits.

What is the statement in italics supposed to mean, and how is the theorem following it supposed to be some sort of weakening of the statement? I cannot seem to get a concrete answer in my head.

The problem is that you cannot choose a domain and codomain for such a putative adjunction consistently and simultaneously. The statement that we have is that the category of presheaves on $C$ is the free cocomplete category on $C$ when $C$ is small. However, the forgetful functor from cocomplete categories to categories does not land in small categories-every cocomplete category which is not a preorder is large.

So, you might want an improved adjunction between large categories and cocomplete categories. However, the category of presheaves on a large category is even larger than large! What this means depends on your foundations. If we work with universes so that a small category is $U_1$-small, then presheaves on a small category are small with respect to the next biggest universe $U_2$. Now presheaves on a $U_2$-small category only have the appropriate universal property if we can them to be presheaves of $U_2$-small sets, and such presheaves are not $U_2$-small.

So the presheaves-forgetful functor pair cannot form an adjunction, because the desired left adjoint moves us up a universe every time we apply it. Thus in particular we cannot get around this straightforwardly by using universes.

There are a couple of partial solutions to this problem. The simplest is to regard the formation of the presheaf category as a relative left adjoint to the forgetful functor from cocomplete categories to (possibly) large categories. In other words, it behaves like a left adjoint, but is only partially defined. This is a rephrasing of the theorem you quote-the formation of presheaves behaves like a left adjoint when its input is small.

A more technical approach is to ask the question: even if the formation of presheaves cannot be left adjoint to the forgetful functor from cocomplete categories to large categories, does this forgetful functor have any left adjoint at all? In fact it does; for instance, it satisfies a 2-categorical version of the general adjoint functor theorem. This left adjoint sends a category $K$ to the subcategory of presheaves on $K$ formed by the colimits of small diagrams of representable presheaves. However, this category of small presheaves is not nearly as well behaved as the presheaf category. It needn't even be a topos in general.

So to summarize, this is a real issue which cannot be eradicated by any level of generous assumptions on the foundations. It's the go-to example of why size issues cannot be completely ignored in category theory.

It's tempting to say that we have an adjunction $mathrmPsh dashv U$ between functors $mathbfCat rightleftarrows mathbfCat_textcolim$, where $mathbfCat_textcolim$ is the category of all cocomplete categories and cocontinuous functors. But in order to make sense of the definition of 'adjunction', we need to refer to hom sets, meaning that $mathbfCat$ and $mathbfCat_mathrmcolim$ to be locally small.

So really $mathbfCat$ is the category of all small categories, and likewise for $mathbfCat_mathrmcolim$.

However, even if we restricted the statement to small cocomplete categories $mathcalD$, the category $mathrmPsh(mathcalC)$ is not small in general, and so $mathrmPsh$ is not a functor $mathbfCat to mathbfCat_mathrmcolim$.

However, for fixed small $mathcalC$ and locally small cocomplete $mathcalD$, it makes sense to ask whether there is a bijection between the respective hom sets (or, in this case, an equivalence between hom categories), without requiring the constructions $mathrmPsh$ and $U$ to be functors between categories. So that's the result that is proved in your notes.

You could hack it using a hierarchy of Grothendieck universes, but I wouldn't recommend it.