# What do we call the function in this diagram?

#### navigator

We call the function f' in this diagrams:

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

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

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

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

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#### 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.

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