Is bijection a transitive relationship?

Question

As doing some cardinality proofs I began to wonder if bijection is a transitive relationship. For simplicity I tried to use real functions to explore the question.

Consider the bijective functions $f: A \rightarrow B$ and $ g: B \rightarrow C$ is it a necessity that exits a bijective $h: A \rightarrow C$ ?

I tried composing functions:

If $$ \forall b \in B, \exists ! a \in A | f(a) = b $$ $$ \forall c \in C, \exists ! B \in B | g(b) = c $$

Then $$ g(b) = c \iff g(f(a)) = c $$ Defining $ h:= g \circ f$, then h is a bijection $A \rightarrow C$

I have three questions:

1 - What I tried was correct?

2 - If so, then is there any way to generalize it?

3 - Is bijection a transitive relationship?

Answer 1

1. Yes, what you tried was correct.
2. Generalize it how? A bijection is an example of an equivalence relation between sets.
3. Yes, for exactly the reasons you mentioned.