Is it consistent in $\mathsf{ZF}$ that there is an infinite cardinal $\kappa$, cardinals $\alpha, \beta\in\kappa$ and a function $f:\kappa\to \alpha$ such that for each $x\in\alpha$ there is an injective map $i:f^{-1}(\{x\})\to \beta$?



Function $f:\kappa\to\alpha$ with small fibers where $\alpha\in\kappa$
Yes, this is consistent. 


Consider a (transitive) model of $\mathrm{ZF}$ in which $\omega\_1$ has countable cofinality. Fix a strictly increasing, cofinal sequence $(\xi\_n \mid n < \omega)$ in $\omega\_1$. Consider
$$
f \colon \omega\_1 \to \omega, x \mapsto \min \{n < \omega \mid x < \xi\_n \}.
$$
$f^{-1}(\{n\})$ is bounded in $\omega\_1$ for all $n \in \omega$. Hence there is an injection
$$
i \colon f^{-1}(\{n\}) \to \omega.
$$


