Question about if $g\circ f$ is surjective then $g$ is surjective. Suppose we have $f:A\rightarrow B$ and $g:B\rightarrow C$
if $g\circ f$ is surjective then $g$ is surjective.
This seems one directional as in $\Rightarrow$, but my Professor said if it is a definition then it is always an $\iff$ no matter how the statement is said (or that is how I am interpreting his words, correct me if I am wrong!)
Proof:
Let $c\in C$ be arbritrary. Given that $g\circ f$ is surjective, there exists an $a\in A$ such that $g\circ f(a)= g(f(a))=c$. This implies we have some $b=f(a)\in B$ such that $g(b)= c$. Hence we have shown $g$ is surjective.
But I am wondering is it necessarily true if we try to prove the backwards direction? I seem to run into the problem that I am uncertain if $B$ in $f$ is the same $B$ in $g$.
 A: The backwards direction isn't true in general. Here's a counter example.
let $A=B=C=\mathbb{R}$
let $f(x)=1$.
let $g(x)=x$.
$g(x)$ is surjective.
$f(g(x)) = 1$ which is not surjective
There must have been something he said wrong, or you misunderstood.
EDIT: I agree with peek-a-boo. Is it possible he was describing 'if and only if"? "if and only if" does correspond to definitions.
A: Your proof is correct.  The converse is not true.  For example, $g: \{1,2\} \rightarrow \{1,2\}, 1 \mapsto 1, 2\mapsto 2$ is surjective, but if $f: \{1\} \rightarrow \{1\}$ is defined by $1 \mapsto 1$ then $g \circ f$ is not surjective.
Probably what your professor meant was that when we define something, we sometimes use "if" and let "only if" be implied. e.g.,

We say that $f:A\rightarrow B$ is surjective if for all $b \in B$ there exists some $a \in A$ with $f(a) = b$.

This is a definition, so really "if" means "if and only if".  The reason is that a definition must be an "if and only if" condition to be sensible; otherwise it is ill-defined.  Context clues tell you it is a definition: the use of "we say that" and italics around the term surjective (and perhaps that the term has not been used before in the text.)  In your case, the statement is not a definition, so the converse is not implied.
