Moschovakis, (part of) Exercise x4.16:
Prove that for all cardinal numbers $\kappa,\lambda,\mu$ $$\lambda\le_c\mu\implies \kappa^\lambda\le_c\kappa^\mu$$ provided $\kappa\ne 0$.
$\le_c$ means "less than or equal in size"; $(A\to B)$ denotes the set of functions from $A$ to $B$.
So we are given an injection $\lambda\to \mu$. It suffices to construct an injection $(\lambda\to\kappa)\to(\mu\to\kappa)$.
I think the idea should be like this. If $l\mapsto m$ under the given injection then we could define the map we need by sending the map $\lambda\to\kappa,l\mapsto k$ to the map $\mu\to\kappa,m\mapsto k$.
My concern is in organizing this notationally. Is it okay to just say what I said? The potential problem is that a map $\lambda\to \mu$ is given by some data that sends every element of $\lambda$ to an element of $\mu$, but I've specified only where the particular element $l$ goes to. The similar problem arises for the other maps. Moreover, I don't see how to prove injectiveness in this notation.
And why do we need $\kappa\ne 0$? (Recall $0$ is defined as $0=\emptyset=|\emptyset|$) I think in this case the set $(\lambda\to \kappa)$ is empty when $\lambda=\emptyset$. In such a case, there is the empty function which is injective. If $\lambda\ne\emptyset$ then $(\lambda\to\kappa)$ is a singleton but $(\mu\to\kappa)$ is empty provided $\mu\ne \emptyset$. So the only occasion when the implication in question fails (under $\kappa =0$) is when $\lambda=\emptyset, \mu\ne \emptyset$?