I am a beginner of Algebraic topology course. I just come across the definition of Homotopy
Homotopy of 2 maps:
$f,g:S\to T$ are 2 continuous maps is said to called homotopic if there exist continuous map $H:S\times [0,1]\to T$ such that $H(s,0)=f(s)$ and $H(s,1)=g(s)$.
Homotopy of 2 spaces :
Two topological spaces $X,Y$ are said to homotopic if there is continous map $f:S\to T$ and $g:T\to S$ such that $f\circ g:T\to T$ is homotopic to Identity on T and $g\circ f:S\to S$ is homotopic to Identity on S.
I know that to define second definition we used the first one.Also intuitively for the first defination, if we consider graphs of 2 function then one continuously deformed to another
Is this continuous deformation happen in 2nd definition .But how to interpret that from definition only?
This question arises when I am reading and try to relate both definition
Please Help me