Let E be a Banach space with 1-unconditional basis. Denote by $\Delta \left(\otimes {̂}_{n,\pi }E\right)$ (resp. $\Delta \left(\otimes {̂}_{n,s,\pi }E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by $\Delta \left(\otimes {̂}_{n,\left|\pi \right|}E\right)$ (resp. $\Delta \left(\otimes {̂}_{n,s,\left|\pi \right|}E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to $E_{\left[n\right]}$, the completion of the n-concavification of...