# direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $$A$$ with continuous linear maps $$0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0.$$ We can take inductive limit (as TVS) and get: $$0\to B\;{\xrightarrow {\ f\ }}\;C\;{\xrightarrow {\ g\ }}\;D\to 0.$$ I know, that direct limit preserves exactness. But I would like to know, new maps will remain continuous?

Thank you so much!