Say we have two monoids $N,M$ and w.l.o.g. assume that $|N| \geq |M|$. Does there exist a surjective homomorphism $\varphi : N \to M$?
Context
In the category of sets the answer would be yes: there is a surjection. The question is whether a surjection can me made compatible with the monoid structure.
If you just want a homomorphism, then for any two monoids $M$ and $N$, the function $f:N\to M$ defined by $f(n)=1_M$ where $1_M$ is the identity element of $M$, is a homomorphism. This trivial homomorphism exists no matter what the sizes of $M$ and $N$ are.
However, if you want a surjective homomorphism, the answer is no: For example, there is no non-trivial homomorphism from $\mathbb Z_3$ to $\mathbb Z_2$