Let $T = \bigcup T_i$ be an affine open covering of the scheme $T$. If we can show that the open subspaces $X_i = f^{-1}(T_i)$ are representable, then $X$ is representable, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme}. Note that $X_i = T_i \times_T X$ and that locally quasi-finite and separated are both stable under base change, see Lemmas \ref{lemma-base-change-separated} and \ref{lemma-base-change-quasi-finite}. Hence we may assume $T$ is an affine scheme. \medskip\noindent By Properties of Spaces, Lemma \ref{spaces-properties-lemma-union-of-quasi-compact} there exists a Zariski covering $X = \bigcup X_i$ such that each $X_i$ has a surjective \'etale covering by an affine scheme. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme} again it suffices to prove the proposition for each $X_i$. Hence we may assume there exists an affine scheme $U$ and a surjective \'etale morphism $U \to X$. This reduces us to the situation in the next paragraph. \medskip\noindent Assume we have $$ U \longrightarrow X \longrightarrow T $$ where $U$ and $T$ are affine schemes, $U \to X$ is \'etale surjective, and $X \to T$ is separated and locally quasi-finite. By Lemmas \ref{lemma-etale-locally-quasi-finite} and \ref{lemma-composition-quasi-finite} the morphism $U \to T$ is locally quasi-finite. Since $U$ and $T$ are affine it is quasi-finite. Set $R = U \times_X U$. Then $X = U/R$, see Spaces, Lemma \ref{spaces-lemma-space-presentation}. As $X \to T$ is separated the morphism $R \to U \times_T U$ is a closed immersion, see Lemma \ref{lemma-fibre-product-after-map}. In particular $R$ is an affine scheme also. As $U \to X$ is \'etale the projection morphisms $t, s : R \to U$ are \'etale as well. In particular $s$ and $t$ are quasi-finite, flat and of finite presentation (see Morphisms, Lemmas \ref{morphisms-lemma-etale-locally-quasi-finite}, \ref{morphisms-lemma-etale-flat} and \ref{morphisms-lemma-etale-locally-finite-presentation}). \medskip\noindent Let $(U, R, s, t, c)$ be the groupoid associated to the \'etale equivalence relation $R$ on $U$. Let $u \in U$ be a point, and denote $p \in T$ its image. We are going to use More on Groupoids, Lemma \ref{more-groupoids-lemma-quasi-finite-over-base-j-proper} for the groupoid $(U, R, s, t, c)$ over the scheme $T$ with points $p$ and $u$ as above. By the discussion in the previous paragraph all the assumptions (1) -- (7) of that lemma are satisfied. Hence we get an \'etale neighbourhood $(T', p') \to (T, p)$ and disjoint union decompositions $$ U_{T'} = U' \amalg W, \quad R_{T'} = R' \amalg W' $$ and $u' \in U'$ satisfying conclusions (a), (b), (c), (d), (e), (f), (g), and (h) of the aforementioned More on Groupoids, Lemma \ref{more-groupoids-lemma-quasi-finite-over-base-j-proper}. We may and do assume that $T'$ is affine (after possibly shrinking $T'$). Conclusion (h) implies that $R' = U' \times_{X_{T'}} U'$ with projection mappings identified with the restrictions of $s'$ and $t'$. Thus $(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \times_{t', U', s'} R'})$ of conclusion (g) is an \'etale equivalence relation. By Spaces, Lemma \ref{spaces-lemma-finding-opens} we conclude that $U'/R'$ is an open subspace of $X_{T'}$. By conclusion (d) the schemes $U'$, $R'$ are affine and the morphisms $s'|_{R'}, t'|_{R'}$ are finite \'etale. Hence Groupoids, Proposition \ref{groupoids-proposition-finite-flat-equivalence} kicks in and we see that $U'/R'$ is an affine scheme. \medskip\noindent We conclude that for every pair of points $(u, p)$ as above we can find an \'etale neighbourhood $(T', p') \to (T, p)$ with $\kappa(p) = \kappa(p')$ and a point $u' \in U_{T'}$ mapping to $u$ such that the image $x'$ of $u'$ in $|X_{T'}|$ has an open neighbourhood $V'$ in $X_{T'}$ which is an affine scheme. We apply Lemma \ref{lemma-neighbourhood-scheme} to obtain an open subspace $W \subset X$ which is a scheme, and which contains $x$ (the image of $u$ in $|X|$). Since this works for every $x$ we see that $X$ is a scheme by Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme}. This ends the proof.