The following is from Schaefer, "Topological Vector Spaces", 1999, p. 56/57:
Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex spaces with $\alpha$ in a directed poset $A$ and $h_{\beta \alpha} : E_\alpha \to E_\beta$ continuous linear maps for $\alpha \leq \beta$. Set $F := \oplus_\alpha E_\alpha$ and let $g_\alpha : E_\alpha \to F$ be the canonical imbedding. Denote by $H$ the subspace of $F$ generated by the ranges of all the maps $g_\alpha - g_\beta h_{\beta \alpha} : E_\alpha \to F$ for $\alpha \leq \beta$.
If $H \subseteq F$ is closed then one can represent the inductive limit of the family $E_\alpha$ in the category of Hausdorff locally convex spaces by $F / H$ which is then a Hausdorff locally convex space.
Now Schaefer writes: "It appears to be unknown whether $H$ is necessarily closed in $F$."
It seems that this statement is equivalent to say that the inductive limit does not necessarily exist.
Does one know here whether this is still an open issue?
EDIT: Summarizing two answers: The inductive limit $F / H$ in the cat. of l.c.s. does exist and is not necessarily Hausdorff. Also, the inductive limit in the cat. of Hausdorff l.c.s. does also exist and is equal to $F / \overline{H}$. For closed $H$ they coincide.