Suppose that $\forall j\in J: X_j $ is a locally convex space, with defining family of seminorms $(q_{jk})_{k \in K_j}$. Also let $X$ be a vector space and $T_j: X \to X_j$ a linear map $\forall j\in J$.
Why does the initial topology on X w.r.t. the family of linear maps $(T_j)_j$ coincide with the locally convex topology defined by the seminorms $\{q_{jk} \circ T_j | j \in J, k \in K_j \}$?