What can be said about this map in a commutative diagram?

Question

Suppose that I have the following commutative diagram of maps between sets:

where $u,v$ are bijections. What can be said about the map $\varphi$ in the middle? Is it possible to conclude $\varphi$ is injective, surjective, or bijective?

Thank you.

Answer 1

Nothing can be said. Suppose for simplicity that $X=Y=V=\{0\}$, with $u$ and $v$ being the unique functions $0\mapsto 0$. Then $g$ must be the constant map $z\mapsto 0$ and there is a unique $u_0\in U$ that defines $h$ via $0\mapsto u_0$.

Therefore $\varphi = hg:Z\to U$ is also a constant map $z\mapsto u_0$

It is easy now to adjust the sizes of $U$ and $Z$ to force $\varphi$ to be injective, surjective, both, or neither.

More generally, the bijectivity conditions imply that $g$ is surjective and $h$ is injective, which seems good until you notice that these properties are "in the wrong order". (That is, if a bijection composed of two maps, it's the first map that has to be injective, and the second that has to be surjective.)