Let $X$ be a locally convex space, and $\{Y_i;\ i\in I\}$ a covariant system of locally convex spaces over a partially ordered set $I$, i.e. a system of linear continuous mappings is given $\sigma^j_i:Y_i\to Y_j$, $i\le j\in I$ such that $$ \sigma^k_j\circ\sigma^j_i=\sigma^k_i, \qquad i\le j\le k\in I. $$ (We assume that all locally convex spaces are over a common field, say, $\mathbb C$. If necessary, we can think that $X$ and $Y_i$ are complete.)
Let $Y$ be the injective limit of $\{Y_i;\ i\in I\}$ in the category of locally convex spaces $$ Y=\injlim_{i\in I}Y_i. $$ For each $i\in I$ we consider the space ${\mathcal L}(X,Y_i)$ of operators (i.e. linear continuous mappings) $\varphi:X\to Y_i$ endowed with the compact-open topology. Similarly, we endow the space ${\mathcal L}(X,Y)$ of operators $\varphi:X\to Y$ with the compact-open topology.
The family of spaces $\{{\mathcal L}(X,Y_i);\ i\in I\}$ is also a covariant system over $I$, and we can consider its injective limit in the category of locally convex spaces: $$ \injlim_{i\in I}{\mathcal L}(X,Y_i). $$ There is a natural mapping $$ \omega:\injlim_{i\in I}{\mathcal L}(X,Y_i)\to {\mathcal L}(X,Y). $$
Question:
Is it true that the topology of $\injlim_{i\in I}{\mathcal L}(X,Y_i)$ is inherited from ${\mathcal L}(X,Y)$?
In other words,
if $U$ is a neighbourhood of zero in $\injlim_{i\in I}{\mathcal L}(X,Y_i)$ do there always exist a compact set $T\subseteq X$ and a neighbourhood of zero $V$ in $Y$ such that $$ U\supseteq \omega^{-1}(\{\varphi\in{\mathcal L}(X,Y):\ \varphi(T)\subseteq V\} ) $$ ?
