I am currently reading this paper on derived blow up, in definition 2.4.1, I am faced with such situation:
if we denote the infinity category of simplicial ring as $Alg$ and the 1 category of commutative monoids as $Mon$. And we denote the infinity category having objects as presentable infinity categories and morphisms between them the left adjoint functors as $Pr^{L}$. In the paper, we already have a functor $Alg_{(-)}^{(-)}: Alg\times Mon\rightarrow Pr^{L}$. By HTT, proposition 5.5.3.13, $Pr^{L}$ admits all $\textbf{small}$ limits, then the author says we have a right Kan extension along the embedding $Alg\times Mon\rightarrow algSt^{op}\times Mon$ where $algSt$ is the infinite category of algebraic (derived) stack.
In the classical case, for $K:A\rightarrow B$ and $F:A\rightarrow C$, right Kan extension of $F$ along $K$ is computed objectwise by the following formula: $Kan^{R}(F)(b)=limit(b\downarrow K\rightarrow A\rightarrow C)$ for $b\in B$. So we have the conclusion: if $A$ is $\textbf{small}$ and $C$ is $\textbf{complete}$, then right Kan extension in such case exists (here $b\downarrow K$ is the comma category).
Now let's turn to the infinity category: $Alg=Fun(Poly^{op},Spc)$ which is presentable by definition, and the proposition 5.5.3.13 in HTT only states that $Pr^{L}$ has $\textbf{small}$ limits, so to make sure that our old method comes into effect, we need to make sure the limit to compute the pointwise Kan extension value is $\textbf{small}$, which is in my eyes hard to see, because $Alg$ is far from small and the collection of all morphisms $Spec(R)\rightarrow X$ for an algebraic stack $X$ is unfathomable, so how do we solve this problem?
Also such questions on the existence of left Kan extension along some embeddings appear on many occasions, for example, the infinity category of quasi-coherent sheaf on the derived scheme is defined to be the Kan extension along the derived affine schemes's embedding into the derived schemes. It seems that $\textbf{the size issue}$ is not a problem at all and we only need to consider whether the codomain admits small (co)limits. I am very confused why people seem to feel at ease to formulate a Kan extension along the embedding in derived algebraic geometry and never worry about size issue.
