The Stacks project

Remark 76.17.7. Another special case of Lemmas 76.17.1, 76.17.2, 76.17.4, and 76.17.5 is where $B$ itself is a thickening $Z \subset Z' = B$ and $Y = Z \times _{Z'} Y'$. Picture

In this case the map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ is determined by $a$: the map $h^*\mathcal{C}_{Z/Z'} \to \mathcal{C}_{Y/Y'}$ is surjective (because we assumed $Y = Z \times _{Z'} Y'$), hence the pullback $g^*\mathcal{C}_{Z/Z'} = a^*h^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'}$ is surjective, and the composition $g^*\mathcal{C}_{Z/Z'} \to a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ has to be the canonical map induced by $g'$. Thus the sheaf of Lemma 76.17.4 is just given by the rule

and we act on this by the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(a^*\Omega _{Y/Z}, \mathcal{C}_{X/X'})$.