$\nabla_{i,j}$ as follows: first, we have the commutative square $$\xymatrix{ I_{i+1}\wedge I_j \ar[r]^{\rho_{i+1,j}} \ar[d]^{f_{i+1}\wedge \id_{I_j}} & I_{i+j+1}\ar[d]^{f_{i+j+1}} \\
    I_i\wedge I_j \ar[r]^{\rho_{i,j}} & I_{i+j} }$$ so, using the assumption that the maps $f_i$ are cofibrations, we take vertical cofibers to get a map $$\tilde{\nabla}_{i,j}: I_i/I_{i+1}\wedge I_j\rightarrow I_{i+j}/I_{i+j+1} .$$

    Now we have the commutative diagram $$\xymatrix{ I_{i+1}\wedge I_{j+1}\ar[rd]^{\id_{I_{i+1}}\wedge f_{j+1}}\ar[dd]^{f_{i+1}\wedge \id_{I_{j+1}}}\ar[rr]^{\rho_{i+1,j+1}} & & I_{i+j+2}\ar[rd]^{f_{i+j+2}}\ar[dd]^(.7){f_{i+j+2}} & \\
     & I_{i+1}\wedge I_j \ar[dd]^(.7){f_{i+1}\wedge \id_{I_j}} \ar[rr]^(.35){\rho_{i+1,j}} & & I_{i+j+1}\ar[dd]^{f_{i+j+1}} \\
    I_i\wedge I_{j+1}\ar[rd]^{\id_{I_i}\wedge f_{j+1}} \ar[rr]^(.4){\rho_{i,j+1}} \ar[dd] & & I_{i+j+1}\ar[rd]^{f_{i+j+1}}\ar[dd] & \\
     & I_i\wedge I_j\ar[rr]^(.4){\rho_{i,j}} \ar[dd] & & I_{i+j} \ar[dd] \\
    I_i/I_{i+1}\wedge I_{j+1} \ar[rd]_{\id_{I_i/I_{i+1}}\wedge f_{j+1}} \ar[rr]^(.4){\tilde{\nabla}_{i,j+1}} & & I_{i+j+1}/I_{i+j+2}\ar[rd]^{0} & \\
     & I_i/I_{i+1}\wedge I_j \ar[rr]^{\tilde{\nabla}_{i,j}} & & I_{i+j}/I_{i+j+1} }$$ in which the columns are cofiber sequences. We therefore have a choice of factorization of the composite map $\tilde{\nabla}_{i,j}\circ \left( \id_{I_i/I_{i+1}}\wedge f_{j+1}\right)$ through the zero object. Thus, we have the commutative square $$\xymatrix{ I_i/I_{i+1}\wedge I_{j+1}\ar[d]^{\id_{I_i/I_{i+1}}\wedge f_{j+1}} \ar[r] & 0 \ar[d] \\
    I_i/I_{i+1}\wedge I_j \ar[r]^{\tilde{\nabla}_{i,j}} & I_{i+j}/I_{i+j+1} }$$ and, taking vertical cofibers, a map $$I_i/I_{i+1}\wedge I_j/I_{j+1}\rightarrow I_{i+j}/I_{i+j+1},$$ which we compose with the inclusion map $I_{i+j}/I_{i+j+1}\hookrightarrow E_0^*I $ to produce our desired map $$\nabla_{i,j}: I_i/I_{i+1}\wedge I_j/I_{j+1}\rightarrow E_0^*I .$$

Let $S$ be a finite set. We can equip $\mathbb{N}^S$ with the $L_1$ norm 