For an ordinal $\alpha $, we set $\text{Cov}(\mathcal{C})_\alpha = \text{Cov}(\mathcal{C}) \cap V_\alpha $. Given an ordinal $\alpha $ and a cardinal $\kappa $, we set $\text{Cov}(\mathcal{C})_{\kappa , \alpha }$ equal to the set of elements $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_\alpha $ such that $|I| \leq \kappa $.

**Proof.**
To prove this, we first consider the set $\mathcal{S}$ of all sets of morphisms of $\mathcal{C}$ with fixed target. In other words, an element of $\mathcal{S}$ is a subset $T$ of $\text{Arrows}(\mathcal{C})$ such that all elements of $T$ have the same target. Given a family $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$ of morphisms with fixed target, we define $Supp(\mathcal{U}) = \{ \varphi \in \text{Arrows}(\mathcal{C}) \mid \exists i\in I, \varphi = \varphi _ i\} $. Note that two families $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$ and $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ are combinatorially equivalent if and only if $Supp(\mathcal{U}) = Supp(\mathcal{V})$. Next, we define $\mathcal{S}_\tau \subset \mathcal{S}$ to be the subset $\mathcal{S}_\tau = \{ T \in \mathcal{S} \mid \exists \ \mathcal{U} \in \text{Cov}(\mathcal{C}) \ T = Supp(\mathcal{U})\} $. For every element $T \in \mathcal{S}_\tau $, set $\beta (T)$ to equal the least ordinal $\beta $ such that there exists a $\mathcal{U} \in \text{Cov}(\mathcal{C})_\beta $ such that $T = \text{Supp}(\mathcal{U})$. Finally, set $\beta _0 = \sup _{T \in S_\tau } \beta (T)$. At this point it follows that every $\mathcal{U} \in \text{Cov}(\mathcal{C})$ is combinatorially equivalent to some element of $\text{Cov}(\mathcal{C})_{\beta _0}$.

Let $\kappa $ be the maximum of $\aleph _0$, the cardinality $|\text{Arrows}(\mathcal{C})|$,

\[ \sup \nolimits _{\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_{\beta _0}} |I|, \quad \text{and}\quad \sup \nolimits _{\{ U_ i \to U\} _{i\in I} \in \text{Cov}_0} |I|. \]

Since $\kappa $ is an infinite cardinal, we have $\kappa \otimes \kappa = \kappa $. Note that obviously $\text{Cov}(\mathcal{C})_{\beta _0} = \text{Cov}(\mathcal{C})_{\kappa , \beta _0}$.

We define, by transfinite induction, a function $f$ which associates to every ordinal an ordinal as follows. Let $f(0) = 0$. Given $f(\alpha )$, we define $f(\alpha + 1)$ to be the least ordinal $\beta $ such that the following hold:

We have $\alpha + 1 \leq \beta $ and $f(\alpha ) \leq \beta $.

If $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa , f(\alpha )}$ and for each $i$ we have $\{ W_{ij} \to U_ i\} _{j\in J_ i} \in \text{Cov}(\mathcal{C})_{\kappa , f(\alpha )}$, then $\{ W_{ij} \to U\} _{i \in I, j\in J_ i} \in \text{Cov}(\mathcal{C})_{\kappa , \beta }$.

If $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa , \alpha }$ and $W \to U$ is a morphism of $\mathcal{C}$, then $\{ U_ i \times _ U W \to W \} _{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa , \beta }$.

To see $\beta $ exists we note that clearly the collection of all coverings $\{ W_{ij} \to U\} $ and $\{ U_ i \times _ U W \to W \} $ that occur in (2) and (3) form a set. Hence there is some ordinal $\beta $ such that $V_\beta $ contains all of these coverings. Moreover, the index set of the covering $\{ W_{ij} \to U\} $ has cardinality $\sum _{i \in I} |J_ i| \leq \kappa \otimes \kappa = \kappa $, and hence these coverings are contained in $\text{Cov}(\mathcal{C})_{\kappa , \beta }$. Since every nonempty collection of ordinals has a least element we see that $f(\alpha + 1)$ is well defined. Finally, if $\alpha $ is a limit ordinal, then we set $f(\alpha ) = \sup _{\alpha ' < \alpha } f(\alpha ')$.

Pick an ordinal $\beta _1$ such that $\text{Arrows}(\mathcal{C}) \subset V_{\beta _1}$, $\text{Cov}_0 \subset V_{\beta _0}$, and $\beta _1 \geq \beta _0$. By construction $f(\beta _1) \geq \beta _1$ and we see that the same properties hold for $V_{f(\beta _1)}$. Moreover, as $f$ is nondecreasing this remains true for any $\beta \geq \beta _1$. Next, choose any ordinal $\beta _2 > \beta _1$ with cofinality $\text{cf}(\beta _2) > \kappa $. This is possible since the cofinality of ordinals gets arbitrarily large, see Proposition 3.7.2. We claim that the pair $\kappa $, $\alpha = f(\beta _2)$ is a solution to the problem posed in the lemma.

The first and third property of the lemma holds by our choices of $\kappa $, $\beta _2 > \beta _1 > \beta _0$ above.

Since $\beta _2$ is a limit ordinal (as its cofinality is infinite) we get $f(\beta _2) = \sup _{\beta < \beta _2} f(\beta )$. Hence $\{ f(\beta ) \mid \beta < \beta _2\} \subset f(\beta _2)$ is a cofinal subset. Hence we see that

\[ V_\alpha = V_{f(\beta _2)} = \bigcup \nolimits _{\beta < \beta _2} V_{f(\beta )}. \]

Now, let $\mathcal{U} \in \text{Cov}_{\kappa , \alpha }$. We define $\beta (\mathcal{U})$ to be the least ordinal $\beta $ such that $\mathcal{U} \in \text{Cov}_{\kappa , f(\beta )}$. By the above we see that always $\beta (\mathcal{U}) < \beta _2$.

We have to show properties (1), (2), and (3) defining a site hold for the pair $(\mathcal{C}, \text{Cov}_{\kappa , \alpha })$. The first holds because by our choice of $\beta _2$ all arrows of $\mathcal{C}$ are contained in $V_{f(\beta _2)}$. For the third, we use that given a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I} \in \text{Cov}(\mathcal{C})_{\kappa , \alpha }$ we have $\beta (\mathcal{U}) < \beta _2$ and hence any base change of $\mathcal{U}$ is by construction of $f$ contained in $\text{Cov}(\mathcal{C})_{\kappa , f(\beta + 1)}$ and hence in $\text{Cov}(\mathcal{C})_{\kappa , \alpha }$.

Finally, for the second condition, suppose that $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa , f(\alpha )}$ and for each $i$ we have $\mathcal{W}_ i = \{ W_{ij} \to U_ i\} _{j\in J_ i} \in \text{Cov}(\mathcal{C})_{\kappa , f(\alpha )}$. Consider the function $I \to \beta _2$, $i \mapsto \beta (\mathcal{W}_ i)$. Since the cofinality of $\beta _2$ is $> \kappa \geq |I|$ the image of this function cannot be a cofinal subset. Hence there exists a $\beta < \beta _1$ such that $\mathcal{W}_ i \in \text{Cov}_{\kappa , f(\beta )}$ for all $i \in I$. It follows that the covering $\{ W_{ij} \to U\} _{i\in I, j \in J_ i}$ is an element of $\text{Cov}(\mathcal{C})_{\kappa , f(\beta + 1)} \subset \text{Cov}(\mathcal{C})_{\kappa , \alpha }$ as desired.
$\square$

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