Condition on composition functions for the given fact to be true in case of codomain off is not same the domain of g

Question

We know that for $f:Y \to Z$ and $g:X \to Y$ , if $f \circ g$ is either bijective or just surjective or injective we have $g$ is injective and $f$ is surjective for first case, similarily $f$ is surjective only for second case and $g$ is injective for third case. But what happens when we have functions of form $f:X \to Y$ and $g:Z \to W$ here can we say the same conclusions for $f \circ g$ when its either of the above three cases?

Answer 1

I'm pretty sure the latter case is not valid. The composition of functions is defined by: $$f \circ g, g: X \rightarrow Y, f:Y \rightarrow Z$$ where X, Y, and Z are sets. $\forall$ x $\in$ X, f(x) is in the domain of g. I think that the only case that it would be valid is if Y = Z.