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The answer to the first part is indeed true. In fact, something more general is true. Let $\mathcal{A}$ be a small category and let $\mathcal{C}$ be a cocomplete category (which is locally small, i.e., there is just a set of morphism between any two objects). Then any cocontinuous functor $L \colon \mathrm{Set}^{\mathcal{A}^\mathrm{op}} \rightarrow \mathcal{C}$ has a right adjoint, given by $C \mapsto \mathcal{C}(K-,C)$, where $K \colon \mathcal{A} \rightarrow \mathcal{C}$ is the composite of the Yoneda embedding and $L$.

This is for example proved in Kelly's "Basic concepts of enriched category theory", Theorem 4.51. He proves the enriched version of this result, where $\mathrm{Set}$ is replaced by any complete and cocomplete category $\mathcal{V}$. I must say I don't know of a reference that just treats the $\mathrm{Set}$-case.

If the target is no longer the category of presheaves on some small large category, then this failsmight fail. Take for example $\mathcal{D}$ the large discrete category whose objects are sets, and let $F \colon \mathcal{D} \rightarrow \mathrm{Set}$ be the canonical inclusion functor. Then the functor $\mathrm{Set}\rightarrow \mathrm{Set}^{\mathcal{D}}$ which sends a set $X$ to the functor $F\times X$ (i.e., the functor which sends a set $A$ to $A\times X$) is cocontinuous, because $A\times -$ preserves colimits. However, there is a proper class of natural transformations $F \rightarrow F$ (a natural transformation just amounts to choosing an endomorphism of each set with no compatibility conditions), so if this functor had a right adjoint $R$, then we would have a bijection $\mathrm{Set}(\ast,RF) \cong \mathrm{Set}^{\mathcal{D}}(F,F)$, i.e., $RF$ would have to be a proper class. The reason for this failure is of course that $\mathrm{Set}^\mathcal{D}$ is not locally small. Note that this problem doesn't go away when we use universes: the above example would give you an isomorphism between a small set and a large set, i.e., a set outside of the universe.

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The answer to the first part is indeed true. In fact, something more general is true. Let $\mathcal{A}$ be a small category and let $\mathcal{C}$ be a cocomplete category (which is locally small, i.e., there is just a set of morphism between any two objects). Then any cocontinuous functor $L \colon \mathrm{Set}^{\mathcal{A}^\mathrm{op}} \rightarrow \mathcal{C}$ has a right adjoint, given by $C \mapsto \mathcal{C}(K-,C)$, where $K \colon \mathcal{A} \rightarrow \mathcal{C}$ is the composite of the Yoneda embedding and $L$.

This is for example proved in Kelly's "Basic concepts of enriched category theory", Theorem 4.51. He proves the enriched version of this result, where $\mathrm{Set}$ is replaced by any complete and cocomplete category $\mathcal{V}$. I must say I don't know of a reference that just treats the $\mathrm{Set}$-case.

If the target is no longer the category of presheaves on some small category, then this fails. Take for example $\mathcal{D}$ the large discrete category whose objects are sets, and let $F \colon \mathcal{D} \rightarrow \mathrm{Set}$ be the canonical inclusion functor. Then the functor $\mathrm{Set}\rightarrow \mathrm{Set}^{\mathcal{D}}$ which sends a set $X$ to the functor $F\times X$ (i.e., the functor which sends a set $A$ to $A\times X$) is cocontinuous, because $A\times -$ preserves colimits. However, there is a proper class of natural transformations $F \rightarrow F$ (a natural transformation just amounts to choosing an endomorphism of each set with no compatibility conditions), so if this functor had a right adjoint $R$, then we would have a bijection $\mathrm{Set}(\ast,RF) \cong \mathrm{Set}^{\mathcal{D}}(F,F)$, i.e., $RF$ would have to be a proper class. The reason for this failure is of course that $\mathrm{Set}^\mathcal{D}$ is not locally small. Note that this problem doesn't go away when we use universes: the above example would give you an isomorphism between a small set and a large set, i.e., a set outside of the universe.