Let $T$ be a continuous linear map from between metrizable locally convex spaces $E$ and $F$. Let $H$ be a closed subspace of $F$ such that $TE\subset H$. Then we can also view $T$ as a map from $E$ into $H$. Let $T$ in fact denote the latter, while the map into $F$ be $JT$, where $J$ is the inclusion operator from $H$ into $F$.
How are $T^{**}$ and $(JT)^{**}$ related?
Obviously, $(JT)^{**}=J^{**}T^{**}$, so the question is reduced to what is $J^{**}$. If $F$ is a normed space, then $J^{*}$ is the quotient map from $F^{*}$ onto $F^*/ H^{\bot} $, and then its adjoint is the isometry from $(F^*/ H^{\bot})^*=H^{\bot\bot}=\overline{H}^{w*} $. Thus, in this case $J^{**}$ is the inclusion from $\overline{H}^{w*} $ into $F^{**}$ (and then $T^{**}$ and $(JT)^{**}$ are related in the "expected" way), but can we cay the same in the more general case?
The transposed of $J:H\to Y$, i.e., the restriction map $Y^\ast\to H^\ast$, is surjective by Hahn-Banach but need not be open for the strong topologies (of uniform convergence on all bounded sets). For Frechet spaces, this is related to distinguishedness: A Frechet space $H$ is distinguished (by definition, this means that its strong dual is barrelled or -- by a theorem of Grothendieck -- equivalently: bornological) if and only if $J^*:Y^*\to H^*$ is open for all embeddings $J:H\to Y$. This is due to Palamodov and can also be seen in my Springer Lecture Notes Derived Functors in Functional Analysis, Theorem 7.2.
If $J^*$ isn't open, the dual of $H^*$ might be strictly smaller than $\left(F^\ast/H^\perp\right)^\ast$ (and I think this really happens for some non-distinguished Frechet spaces $H$).