Map between conormal sheaves 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  $$   p ^ { *  }  \mathcal{C} _ { X / Y }  \to  \mathcal{C}  _ { X' / Y' }  $$

 A: 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 $.     
