Is it possible to generate a hash function that produces similar output to SHA-256? Is it possible to create a hash function that generates a correlated to SHA-256 when given the same input?
In other words, given a fixed input X:
SHA-256(X)=A
SHA-256'(X)=B

where he outputs A and B are similar, or correlated.
I was thinking about remodeling a few parts of the SHA-256 function without changing the underlying construction too much. Can someone offer more insight or suggestions on whether it is possible to do this?

EDIT: I was thinking of creating a hash function with a *built-in &quot;backdoor&quot; that allows for preimage attacks. Sort of creating a breakable version of SHA-256.
I was thinking of creating a hash function that produces a similar/correlated output to SHA256, but with an in-built &quot;backdoor&quot; that allows for preimage attacks. Sort of creating a &quot;breakable&quot; version of SHA-256.

That is not possible for arbitrary sized messages and unkeyed hash functions.
Assume that you created a hash function such that $h(m) \approx SHA256(m)$. This similarity is not defined in the question, however, we can consider that it a permutation, or small edit distance, or correlated. Assume that the similarity can be found faster than pre-image attack of SHA256.
Now assume that $h()$ is a weak hash function that you can find pre-images easily. Now;

take $z = SHA256(m)$
use the similarity to find all possible candidates  ( from the premutation, small edit, correlation, or..) $$z_i \in \{ a\;|\; a \text{ similar to } z\}$$
find pre-images of each $z_i$; $$m_i = \operatorname{pre-image-h}(z_i)$$
check all $m_i$ on SHA256 to see that one of them is the pre-image $$ z \stackrel{?}{=} \operatorname{SHA256}(m_i). $$

That is the break of the pre-image of SHA256. Not expected!.