[tex]\begin{displaymath}

\begin{xy}

*!C\xybox{

\xymatrix{

{E}\ar[r]^{i} \ar[d]_{f} & {X} \ar[dl]^{f'}\\

{Y} &}}

\end{xy}

\end{displaymath}

[/tex]

the entension of function f;

(i is an inclusion map)

[tex]\begin{displaymath}

\begin{xy}

*!C\xybox{

\xymatrix{

{E}\ar[r]^{i} \ar[dr]_{f'} & {X} \ar[d]^{f}\\

&{Y} }}

\end{xy}

\end{displaymath}

[/tex]

the restriction of function f;

(i is an inclusion map)

[tex]\begin{displaymath}

\begin{xy}

*!C\xybox{

\xymatrix{

&{E}\ar[d]^{p}\\

{X}\ar[ur]^{f'}\ar[r]_{f} & {Y} } }

\end{xy}

\end{displaymath}

[/tex]

the lifting of function f;

then how do we call the function f' in this this diagram:

[tex]\begin{displaymath}

\begin{xy}

*!C\xybox{

\xymatrix{

{X}\ar[r]^{f}\ar[rd]_{f'} & {Y} \ar[d]^{p}\\

&{E} } }

\end{xy}

\end{displaymath}

[/tex]