This isn't really a question.
In subsection 6.3.1 of his book, Qing Liu defines the conormal sheaf and it's properties. While excellently explained, some morphisms in this section are not so obvious to discern on a first read, so I am writing them down here just in case it helps someone looking (or me when I dig into it in future if need be). For completeness, I'll define the necessary terms.
Definition: A map $ f : X \to Y $ is said to be an immersion if $ f $ is the composition of a closed immersion followed by an open immersion. If $ f : X \to Y $ is an immersion, $ i : X \to V $, $ j : V \to Y $ is a way of factorizing $ f $ and $ \mathcal{J} $ is the ideal sheaf on $ V $ defining $ i $, we can define the conormal sheaf $ \mathcal{C} _ { X / Y } $ to be $ i^{*} ( \mathcal{J} / \mathcal{ J } ^ { 2 } ) $.
$\,$Proposition Let $ f : X \to Y $, $ g : Y \to Z $ be immersions of schemes. (a) We have a natural exact sequence $$ f ^{ * } \mathcal{C} _ { Y / Z } \to \mathcal { C } _ { X / Z } \to \mathcal { C } _ {X / Y } \to 0 $$ b) Let $ Y ' \to Y $ be a morphism, and let $ X' := X \times_{Y} Y' $. Let $ p : X ' \to X $ be the natural morphism. Then, there is a natural surjective morphism $$ p ^ { * } \mathcal{C} _ { X / Y } \to \mathcal{C} _ { X' / Y' } $$
Proof: (a) Because $ f $ and $ g $ are immersions, we get the following diagrams
\begin{array}{cccc} & & & & V'' \\ & & &\nearrow & \downarrow \\ & & V & \xrightarrow{} & V' \\ & \nearrow & \downarrow & \nearrow & \downarrow \\X & \xrightarrow{f} & Y & \xrightarrow{g} & Z & \end{array}
Here, the maps $ i : X \to V $,$ j : V \to V'' $, $ \ell : Y \to V' $ are closed immersions and $ V \to Y $, $ V' \to Z $, $ V'' \to V' $ are open immersions. Let $ k = j \circ i : X \to V'' $, and let $ \mathcal{I} , \mathcal{J} $, $ \mathcal{K} $, $ \mathcal{L} $ be the ideal sheaves of $ i , j $, $ k $, $ \ell $ respectively. We have the following exact sequences
$$ 0 \to I \to \mathcal{O } _ { V } \to i_{*} \mathcal{O}_{X} \to 0 , $$ $$ 0 \to J \to \mathcal{O}_{V''} \to j_{*} \mathcal{O}_{V} \to 0 $$ $$ 0 \to \mathcal{K} \to \mathcal{O } _ { V'' } \to k_{*} \mathcal{O}_{X} \to 0 $$
which lead us to the commutative diagram
$$\require{AMScd} \begin{CD} @. @. @. 0 \\ @. @. @. @VVV \\ @. @. @. j_{*} \mathcal{I} \\ @. @. @. @VVV \\ 0 @>>> \mathcal{J} @>>> \mathcal{O}_{V''} @>>> j _{*} \mathcal{O}_{V} @>>> 0 \\ @. @VVV @| @VVV @. \\ 0 @>>> \mathcal{K} @>>> \mathcal{O}_{V''} @>>> \mathcal { k } _{*} \mathcal{O}_{X} @>>> 0 \\ @. @. @. @VVV \\ @. @. @. 0 \end{CD} $$
This gives us a natural exact sequence
$$ 0 \to \mathcal{J} \to \mathcal{K} \to j _{ * } \mathcal { I } \to 0 $$ which gives
$$ \mathcal { J } / \mathcal { J } ^ { 2 } \to \mathcal { K } / \mathcal { K } ^ { 2 } \to j _{ * } \mathcal ( { I } / \mathcal { I } ^ { 2 } ) \to 0 . $$
Pulling back along $ k = j \circ i $, we get $$ \boxed { k ^ { * } \mathcal { J } / \mathcal { J } ^ { 2 } \to \mathcal { C } _ { X / Z } \to k ^ { * } j _{ * } \mathcal { I } / \mathcal { I } ^ { 2 } \to 0 } $$
We claim that $ k ^ { * } ( \mathcal { J } / \mathcal { J } ^ { 2 } ) = f ^ { * } \mathcal { C } _ { Y / Z } $ and $ k ^ { * } j _{ * } ( \mathcal { I } / \mathcal { I } ^ { 2 } ) = \mathcal { C } _ { X / Y } $
For the first, let $ \varphi : V \to Y $ and $ \psi : V'' \to V' $ be the open immersions. Then, we see that $$ f ^ { * } \mathcal { C } _ { Y / Z } = i ^ { * } \varphi ^ { * } \ell ^ { * } ( \mathcal{ L } / \mathcal{ L } ^{2} ) = i ^{ * } j ^ { * } \psi ^ { * } ( \mathcal { L } / \mathcal { L } ^ { 2 } ) = i ^ { * } j ^ { * } ( \ell _{ * } ( \mathcal{L} / \mathcal{L} ^ { 2} ) ) _ { \mid V'' } = k ^ { * } ( \mathcal { L } / \mathcal { L } ^{2} _ { \mid V'' } ) = k ^{ * } ( \mathcal { J } / \mathcal { J } ^ { 2 } ) $$
For the second, note that the natural map $ j ^ { * } j _{ * } \mathcal { I } / \mathcal { I } ^ { 2 } \to \mathcal { I } / \mathcal { I } ^ { 2 } $ is an isomorphism as $ V \to V'' $ is a closed immersion. This in turn induces an isomorphism $ k ^ { * } j _{ * } \mathcal{ I } / \mathcal { I } ^ { 2 } = i ^ { * } j ^ { * } j _ { * } \mathcal { I } / \mathcal { I } ^ { 2 } \to i ^ { * } \mathcal { I } / \mathcal { I } ^ { 2 } = \mathcal { C } _ { X / Y } $.
(b) Let $ q : Y' \to Y $ be the given morphism. Let $ i : X \to V $ and $ i' : X' \to V' = q ^ { * } V $ be the closed immersions that factor $ f : X \to Y $ and $ f' : X' \to Y' $. Then, in a similar manner, we get a natural map $ \mathcal { I } \to \tilde{ q } _{ * } \mathcal { J } $, where $ \tilde{q} : V' \to V = q _ { \mid V' } $, and we just pull this back along the two ways. Surjectivity is the equivalent to the surjectivity of the map $ I \otimes A \to IA $ for an ideal $ I $ of a ring $ A $.