Remark 10.134.5. Let $A \to B$ and $\phi : B \to C$ be ring maps. Then the composition $\mathop{N\! L}\nolimits _{B/A} \to \mathop{N\! L}\nolimits _{C/A} \to \mathop{N\! L}\nolimits _{C/B}$ is homotopy equivalent to zero. Namely, this composition is the functoriality of the naive cotangent complex for the square

$\xymatrix{ B \ar[r]_\phi & C \\ A \ar[r] \ar[u] & B \ar[u] }$

Write $J = \mathop{\mathrm{Ker}}(B[C] \to C)$. An explicit homotopy is given by the map $\Omega _{A[B]/A} \otimes _ A B \to J/J^2$ which maps the basis element $\text{d}[b]$ to the class of $[\phi (b)] - b$ in $J/J^2$.

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