Deterministic outputs based on non-connected inputs
"public shared data" $\:$ = $\:$ the ordered pair whose entries are

[a key for a full-domain statistically injective PRF] and [a signature verification key]

$\hspace{.04 in}f$ is an arbitrary one-way function.

For each positive integer $j$, [$o_j$ is the value of the PRF at $j\hspace{.03 in}$] and [$i_j$ is [[the signature for $o_j$] concatenated with [the string of zeros whose length equals the length of that signature]]]

and $\hspace{.04 in}f_j$ is given by

if $\hspace{.04 in}f_j$'s input has even length and its left half is a valid signature for $o_j$

and its right half is all-zero then output $o_j$ else output $\hspace{.04 in}f$ of $\hspace{.04 in}f_j$'s input.