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 $\text{Cov}(\mathcal{C})_{\beta_0}$. \medskip\noindent 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 $\text{Cov}(\mathcal{C})_{\beta_0} = \text{Cov}(\mathcal{C})_{\kappa, \beta_0}$. \medskip\noindent 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: \begin{enumerate} \item We have $\alpha + 1 \leq \beta$ and $f(\alpha) \leq \beta$. \item 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 $\{W_{ij} \to U\}_{i \in I, j\in J_i} \in \text{Cov}(\mathcal{C})_{\kappa, \beta}$. \item 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 $\{U_i \times_U W \to W \}_{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa, \beta}$. \end{enumerate} 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 $f(\alpha) = \sup_{\alpha' < \alpha} f(\alpha')$. \medskip\noindent 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 \ref{proposition-exist-ordinals-large-cofinality}. We claim that the pair $\kappa$, $\alpha = f(\beta_2)$ is a solution to the problem posed in the lemma. \medskip\noindent The first and third property of the lemma holds by our choices of $\kappa$, $\beta_2 > \beta_1 > \beta_0$ above. \medskip\noindent 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 $\beta(\mathcal{U}) < \beta_2$. \medskip\noindent 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 $\text{Cov}(\mathcal{C})_{\kappa, \alpha}$. \medskip\noindent 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.