We will show that let X and Y be M -finite Banach spaces with canonical M-decompositions $X{cong}{{prod}^{{gamma}_{infty}}_{i=1}}{X^{n_i}}_{i};and;Y{cong}{{prod}^{{ar{gamma}}_{infty}}_{j=1}}{ ilde{Y}^{m_j}}_{j}$, respectively and M and N nonzero locally compact Hausdorff spaces. Then I : $C_{0}$(M,X) ${longrightarrow};C_{0}$(N,Y) is an isometrical isomorphism if and only if r = $ar{r}$ and there are permutation and homeomorphisms and continuous maps such that I = ${I^{-1}}_{N.Y};{circ}I_{w}^{-1}{circ}({{prod}^{gamma}}_{i=1}I_{t_i,u_i}){circ}I_{M,X}$.