The coefficient ideal commutes with smooth pullback

Termwise comparison. For a projection, derivatives in the base directions pull back, and derivatives in the new variables annihilate the pulled-back generators; Leibniz shows that derivatives of their variable-coefficient multiples give no additional generators. The étale comparison and local factorization in [F2] therefore give derivative-ideal equality for every smooth morphism. By [F2] the pullback of the $i$-th summand is ${\phi }^{\ast }({\mathcal{D}}^i(\mathcal{I}),\mu -i)=({\mathcal{D}}^i({\phi }^{\ast }\mathcal{I}),\mu -i)$, and these are exactly the summands of $C({\phi }^{\ast }\mathcal{I},\mu )$.

Summing the summands. Pullback of a sum of marked ideals is the sum of the pullbacks by [F3], so summing the identity of step 1.1 over $0\le i\le \mu -1$ gives ${\phi }^{\ast }C(\mathcal{I},\mu )={\sum }_i({\mathcal{D}}^i({\phi }^{\ast }\mathcal{I}),\mu -i)=C({\phi }^{\ast }\mathcal{I},\mu )$, which is the assertion.