A morphism of pairs $(S, \mathfrak q) \to (S', \mathfrak q')$ is given by an $R$-algebra map $\varphi : S \to S'$ such that $\varphi^{-1}(\mathfrak q') = \mathfrak q$. Let us show that the category of pairs is filtered, see Categories, Definition \ref{categories-definition-directed}. The category contains the pair $(R, \mathfrak p)$ and hence is not empty, which proves part (1) of Categories, Definition \ref{categories-definition-directed}. Suppose that $(S, \mathfrak q)$ and $(S', \mathfrak q')$ are two pairs. Note that $\mathfrak q$, resp.\ $\mathfrak q'$ correspond to primes of the fibre rings $S \otimes \kappa(\mathfrak p)$, resp.\ $S' \otimes \kappa(\mathfrak p)$ with residue fields $\kappa(\mathfrak p)$, hence they correspond to maximal ideals of $S \otimes \kappa(\mathfrak p)$, resp.\ $S' \otimes \kappa(\mathfrak p)$. Set $S'' = S \otimes_R S'$. By the above there exists a unique prime $\mathfrak q'' \subset S''$ lying over $\mathfrak q$ and over $\mathfrak q'$ whose residue field is $\kappa(\mathfrak p)$. The ring map $R \to S''$ is \'etale by Lemma \ref{lemma-etale}. This proves part (2) of Categories, Definition \ref{categories-definition-directed}. Next, suppose that $\varphi, \psi : (S, \mathfrak q) \to (S', \mathfrak q')$ are two morphisms of pairs. Then $\varphi$, $\psi$, and $S' \otimes_R S' \to S'$ are \'etale ring maps by Lemma \ref{lemma-map-between-etale}. Consider $$ S'' = (S' \otimes_{\varphi, S, \psi} S') \otimes_{S' \otimes_R S'} S' $$ Arguing as above (base change of \'etale maps is \'etale, composition of \'etale maps is \'etale) we see that $S''$ is \'etale over $R$. The fibre ring of $S''$ over $\mathfrak p$ is $$ F'' = (F' \otimes_{\varphi, F, \psi} F') \otimes_{F' \otimes_{\kappa(\mathfrak p)} F'} F' $$ where $F', F$ are the fibre rings of $S'$ and $S$. Since $\varphi$ and $\psi$ are morphisms of pairs the map $F' \to \kappa(\mathfrak p)$ corresponding to $\mathfrak p'$ extends to a map $F'' \to \kappa(\mathfrak p)$ and in turn corresponds to a prime ideal $\mathfrak q'' \subset S''$ whose residue field is $\kappa(\mathfrak p)$. The canonical map $S' \to S''$ (using the right most factor for example) is a morphism of pairs $(S', \mathfrak q') \to (S'', \mathfrak q'')$ which equalizes $\varphi$ and $\psi$. This proves part (3) of Categories, Definition \ref{categories-definition-directed}. Hence we conclude that the category is filtered. \medskip\noindent Recall that in the proof of Lemma \ref{lemma-henselization} we constructed $(R_{\mathfrak p})^h$ as the corresponding colimit but starting with $R_{\mathfrak p}$ and its maximal ideal $\mathfrak pR_{\mathfrak p}$. Now, given any pair $(S, \mathfrak q)$ for $(R, \mathfrak p)$ we obtain a pair $(S_{\mathfrak p}, \mathfrak qS_{\mathfrak p})$ for $(R_{\mathfrak p}, \mathfrak pR_{\mathfrak p})$. Moreover, in this situation $$ S_{\mathfrak p} = \colim_{f \in R, f \not \in \mathfrak p} S_f. $$ Hence in order to show the equalities of the lemma, it suffices to show that any pair $(S_{loc}, \mathfrak q_{loc})$ for $(R_{\mathfrak p}, \mathfrak pR_{\mathfrak p})$ is of the form $(S_{\mathfrak p}, \mathfrak qS_{\mathfrak p})$ for some pair $(S, \mathfrak q)$ over $(R, \mathfrak p)$ (some details omitted). This follows from Lemma \ref{lemma-etale}.