f:A→B and
g:B→C are onto.
......... [ Given ]
Then, g∘f:A→C
Let us take an element z in the co-domain.
Now, z is in C and g:B→C is onto.
So, there exists some element y in B, such that,
⇒ g(y)=z ---- ( 1 )
Now, y is in B and f:A→B is onto.
So, there exists some x in A, such that,
⇒ f(x)=y ----- ( 2 )
From ( 1 ) and ( 2 ),
⇒ z=g(y)=g(f(x))=(g∘f)(x)
So, z=(g∘f)(x), where x is in A.
∴ g∘f is onto.