Consider the following statements in $\sf ZF$:

(S) If $A, B$ are nonempty sets, then there is a surjection $s:A \to B$, or there is a surjection $t:B\to A$.

(I) If $A, B$ are sets, then there is a injection $i:A\to B$, or there is a injection $j:B\to A$.

Note that (I) implies (S). Assuming $\sf AC$, both statements are true.

Question. Is there a model of $\sf ZF$ in which (S) holds, but not (I)?

Consider the following statements in $\sf ZF$:

(S) If $A, B$ are nonempty sets, then there is a surjection $s:A \to B$, or there is a surjection $t:B\to A$.

(I) If $A, B$ are sets, then there is an injection $i:A\to B$, or there is an injection $j:B\to A$.

Note that (I) implies (S). Assuming $\sf AC$, both statements are true.

Question. Is there a model of $\sf ZF$ in which (S) holds, but not (I)?