What do we call the function in this diagram?

[navigator]

We call the function f' in this diagrams:

$$\begin{displaymath}<br /> \begin{xy}<br /> *!C\xybox{<br /> \xymatrix{<br /> {E}\ar[r]^{i} \ar[d]_{f} & {X} \ar[dl]^{f'}\\<br /> {Y} &}}<br /> \end{xy}<br /> \end{displaymath}$$
the entension of function f;
(i is an inclusion map)

$$\begin{displaymath}<br /> \begin{xy}<br /> *!C\xybox{<br /> \xymatrix{<br /> {E}\ar[r]^{i} \ar[dr]_{f'} & {X} \ar[d]^{f}\\<br /> &{Y} }}<br /> \end{xy}<br /> \end{displaymath}$$
the restriction of function f;
(i is an inclusion map)

$$\begin{displaymath}<br /> \begin{xy}<br /> *!C\xybox{<br /> \xymatrix{<br /> &{E}\ar[d]^{p}\\<br /> {X}\ar[ur]^{f'}\ar[r]_{f} & {Y} } }<br /> \end{xy}<br /> \end{displaymath}$$
the lifting of function f;

then how do we call the function f' in this this diagram:
$$\begin{displaymath}<br /> \begin{xy}<br /> *!C\xybox{<br /> \xymatrix{<br /> {X}\ar[r]^{f}\ar[rd]_{f'} & {Y} \ar[d]^{p}\\<br /> &{E} } }<br /> \end{xy}<br /> \end{displaymath}$$

 

[navigator]

Oh,the latex codes do not display properly here.
So my question is what is the inverse of lifting? like restriction is the inverse of extension.