What's the proper way of fitting hash digest to encryption scheme? I would like to know what is the proper way of fitting the hash digest to the prime in which the encryption scheme operates. regardless if the bits of the hash digest is larger or smaller than the prime.

I've read that the Cramer-Shoup uses a universal one-way hash function, but didn't state what it is. Wikipedia says its just a property, and  with that I plan on using SHA256. My simulator uses smaller bits for presentation purposes and the larger bits digest of SHA256, I have a problem on how to fit it in. I've read in some forums to use mod on in, is this the proper way ?
A universal one-way hash function (or UOWHF), also known as a target-collision-resistant (or TCR) hash function, is a randomized hash function $H_r(m)$ with the following security: If an adversary commits to a message $m$, then upon being challenged with a random $r$, the adversary cannot find a distinct message $m' \ne m$ such that $H_r(m) = H_r(m')$.  (More details, background, history, and references on UOWHF/TCR, particularly in signature applications.)

Any collision-resistant hash function is obviously also TCR, but TCR is a much weaker security property—much all major ‘cryptographic hash functions’ like SHA-256 including broken ones like MD5 are generally conjectured to exhibit TCR in prefix-hash form $H(r \mathbin\| m)$ and in HMAC form $\operatorname{HMAC-\!}H_r(m)$, but in the off chance that they don't (the Merkle–Damgård construction does not necessarily preserve TCR), there's a generic construction called RMX from Halevi and Krawczyk's research program on randomized signatures, which was standardized by NIST in SP 800-106.  If you like more modern flavors, you could use keyed BLAKE2 or KMAC128 too, since TCR—and the slightly stronger eTCR—was an explicit design goal for SHA-3.

If you want a smaller digest, just truncate the hash function; if you want a larger digest, the easiest way is to use an XOF like the SHA-3 function SHAKE128 or like BLAKE2x.  You could also use SHA-256 in ‘CTR mode’, yielding $H(r \mathbin\| m \mathbin\| 0) \mathbin\| H(r \mathbin\| m \mathbin\| 1) \mathbin\| H(r \mathbin\| m \mathbin\| 2) \mathbin\| \dotsb$, provided you make sure to pad it unambiguously, or use a standard (if somewhat more complicated) construction like HKDF-SHA256 or MGF1 of PKCS#1.