Let $(E_n,i_n)_{n\in\mathbb{N}}$ be an direct system of Banach spaces in the category of locally convex spaces (LCSs) with continuous linear maps and let $E_{\infty}$ by their inductive limit. What is the family of semi-norms defining the LCS topology on $E_{\infty}$?
Similarly, if $(E_n,\pi_n)$ is a projective system of Banach spaces in the category of LCS, then what are the semi-norms defining the topology of their projective limit?
There is a simple abstract description of the semi-norms of an inductive limit of Banach or locally convex space $E_n$ with linking maps $i_n^m:E_n\to E_m$ for $n\le m$ and $i_n^\infty:E_n\to E_\infty$: Just take all semi-norms $p:E_\infty\to[0,\infty)$ such that $p\circ i_n^\infty$ is continuous on $E_n$. In the case of Banach (or normed) spaces $(E_n,\|\cdot\|_n)$ this just means $p\circ i_n^\infty \le c_n\|\cdot\|_n$ for some constants $c_n\ge 0$.
It is quite easy to see that these semi-norms on $E_\infty$ generate a locally convex topology satisfying the universal property of inductive limits, i.e., a linear map $T:E_\infty\to X$ into a locally convex space is continuous if and only if all $T\circ i_n^\infty$ are continuous on $E_n$.
However, If you need to construct a semi-norm, e.g., to apply Hahn-Banach, this abstract description is not very helpful. One can try to make the description more concrete, e.g., also the system of semi-norms $$ p(x)=\inf\left\{\sum_{n=1}^N \varepsilon_n \|x_n\|_n: x_n\in E_n, x=\sum_{n=1}^Nx_n\right\} $$ with $\varepsilon_n>0$ generates the inductive limit topology but, apparently, this is not so easy to handle. There are, however, a number of tricks to make constructions of semi-norms feasible. E.g., compactness of the linking maps helps a lot, then you have so-called LS- or DFS-spaces (duals of Fréchet-Schwartz spaces) or Silva-spaces. Other tricks (not always stating explicitely the relation to locally convex inductive limits) were developped by Malgrange, Ehrenpreis, and Hörmander in distribution theory.
As you also asked for projective limits: This is much easier. Now you have linear maps $\pi_\infty^n:F_\infty\to F_n$ and the system of semi-norms $p_n\circ \pi_\infty^n$ with the given semi-norms on $F_n$ generates the projective limit topology.
