To find Linear transformation by using Transpose of linear transformation

Question

Using linear transformation $T:V \to W$ ,we can induce transpose of a linear transformation $T':W' \to V'$ by using linear functional on W. But how can we induce linear transformation $T:V \to W$ by using transpose of linear transformation $T':W' \to V'$? Actually the original question was to find an isomorphism between L(V,W) and L(W',V').I define isomorphism $\alpha:T \to T^{t}$ but got my doubt while showing onto part.