The Stacks project

Suggested alternative proof:

We can assume that $i$ is a closed immersion by shrinking $X$ if necessary. Let $\mathcal{I}\subset\mathcal{O}_X$ be the ideal sheaf associated to $Z$. Consider the sequence Modules, Lemma 17.28.9, by setting $\mathcal{O}_2\to\mathcal{O}_2'$ equal to $\mathcal{O}_X\to\mathcal{O}_X/\mathcal{I}$ and $\mathcal{O}_1=f^{-1}\mathcal{O}_S$, where $f:X\to S$ is the structure morphism. Since $i^*$ is right exact, we can pull back along $i$ and get an exact sequence $$\mathcal{C}_{Z/X} \to i^*\Omega_{X/S}\otimes_{\mathcal{O}_Z}\mathcal{O}_Z \to i^*\Omega_{(\mathcal{O}_X/\mathcal{I})/f^{-1}\mathcal{O}_S} \to 0,$$ where in the second term we have used Modules, Lemma 17.16.4. Hence, the second term is $i^*\Omega_{X/S}$. On the other hand, by Modules, Lemma 17.28.6, the third term equals $$i^{-1}\Omega_{(\mathcal{O}_X/\mathcal{I})/f^{-1}\mathcal{O}_S}\otimes_{i^{-1}\mathcal{O}_X}\mathcal{O}_Z \cong\Omega_{Z/S}\otimes_{i^{-1}\mathcal{O}_X}\mathcal{O}_Z\cong\Omega_{Z/S},$$ where the last step is done using formula $\mathcal{F}\otimes_{\mathcal{O}_Y}\mathcal{O}_Y/\mathcal{J}\cong\mathcal{F}/\mathcal{J}\mathcal{F}$, for $(Y,\mathcal{O}_Y)$ a ringed space, $\mathcal{J}\subset\mathcal{O}_Y$ an ideal sheaf and $\mathcal{F}$ an $\mathcal{O}_Y$-module.

Okay, I forgot to justify the last claim from the statement in the alternative proof: one sees that the resulting pulled-back map $i^*\Omega_{X/S}\to\Omega_{Z/S}$ equals $c_i$ (Lemma 29.32.8) by applying Modules, Lemma 17.28.13, taking $S=S'=S''$ and $X''\to X'\to X$ to be the morphisms of ringed spaces $(Z,\mathcal{O}_Z)\to (X,\mathcal{O}_X/\mathcal{I})\to(X,\mathcal{O}_X)$.

I'm not sure if any of this is any better the already existing proof. I just felt it was nice to connect Lemma 29.32 to the similar-looking Modules, Lemma 17.28.9.

Last thing: maybe one could make explicit in the statement that the maps are $\mathcal{O}_Z$-linear.

Going to leave as is. There is obviously also some kind of version of this for just ringed spaces under suitable hypotheses.