$\require{HTML}$
This question was inspired by Is iteratively applying the Yoneda embedding interesting?, which apparently had a negative answer due to size issues.
We work in $\mathsf{ZFC+GU}$ ($\mathsf{GU}$ is Grothendieck's universe axiom: for every cardinal, there is a larger strongly inaccessible cardinal). All categories are set-sized.
Given a category $\mathcal{C}$ of size at most $\alpha$, let $\beta > \alpha$ be a strongly inaccessible cardinal. Let $\mathsf{PSh}_{\beta}(\mathcal{C}) = \operatorname{Fun}(\mathcal{C}^{\operatorname{op}}, \mathsf{Set}_{\beta})$. We have the Yoneda embedding $$ Y_{\mathcal{C}, \beta}: \mathcal{C} \hookrightarrow \mathsf{PSh}_{\beta}(\mathcal{C}) $$ Here, $\mathsf{Set}_{\beta}$ denotes the category of sets with cardinality $< \beta$.
Let $\mathcal{C}_1$ be a category of size $\alpha_1$. Let $\beta_1 > \alpha_1$ be a strongly inaccessible cardinal. Set $\mathcal{C}_2 = \mathsf{PSh}_{\beta_1}(\mathcal{C}_1)$. Take the Yoneda embedding $$ Y_{\mathcal{C}_1, \beta_1}: \mathcal{C}_1 \hookrightarrow \mathcal{C}_2 $$ Say $\mathcal{C}_2$ has size at most $\alpha_2$. Let $\beta_2 > \alpha_2$ be a strongly inaccessible cardinal. Set $\mathcal{C}_3 = \mathsf{PSh}_{\beta_2}(\mathcal{C}_2)$. Take the Yoneda embedding $$ Y_{\mathcal{C}_2, \beta_2}: \mathcal{C}_2 \hookrightarrow \mathcal{C}_3 $$ ...
Let $\gamma = \underset{n}{\operatorname{sup}} \beta_n$ and let $\delta > \gamma$ be a strongly inaccessible cardinal. All of the categories involved are of size $< \delta$. Now consider the following diagram in $\mathsf{Cat}_{\delta}$: $$ \mathcal{C}_1 \hookrightarrow \mathcal{C}_2 \hookrightarrow \mathcal{C}_3 \hookrightarrow \ldots $$ Since $|\mathbb{N}| < \delta$, $|\mathbb{N}|$-sized colimits exist in $\mathsf{Cat}_{\delta}$. Hence we have a colimit of this diagram: $$ \begin{array}{cccccc} \mathcal{C}_1 & \hookrightarrow & \mathcal{C}_2 & \hookrightarrow & \mathcal{C}_3 & \hookrightarrow & \ldots \\ \style{display: inline-block; transform: rotate(90deg)}{\hookrightarrow} \rule{0px}{1em} & \style{display: inline-block; transform: rotate(135deg)}{\hookrightarrow} & \style{display: inline-block; transform: rotate(155deg)}{\hookrightarrow} & \ldots \\ \mathcal{D} \\ \end{array} $$
My questions are:
Is $\mathcal{D}$ $\delta$-cocomplete? (i.e. do colimits of size $< \delta$ exist in $\mathcal{D}$)
Is the functor $\mathcal{C}_1 \hookrightarrow \mathcal{D}$ fully faithful?
