# Can proxy re-encrypted cyphertext be identical to encrypted text by the target?

I have the following basic set-up for my question, I am getting to learn about proxy re-encryption and so far what I have looked at in more detail is this re-encryption algorithm from IronCore Labs and their Rust library. It is a multi hop re-encryption scheme (re-encryption steps can be chained), but I do not care so much about this property.

Alice and Bob each have their own public-private key pair and a unidirectional re-encryption algorithm. Alice encrypts $$m$$ with her $$pb_a$$ and obtains $$C_a$$. She also creates a re-encryption key $$r_{a-to-b}$$ using her $$sk_a$$ and $$pb_b$$ of Bob (Bob's public key).

After applying the re-encryption step on $$C_a$$, using $$r_{a-to-b}$$, we would obtain $$C_a'$$.

Bob also knows the message $$m$$ and he obtains $$C_b$$.

Question: Does or can $$C_a' == C_b$$ be a true statement?

Where I want to get with this: Can we check publicly that a re-encryption key is valid and transforms cyphertext from a key pair to another? Without using any of the secret keys, only the re-encryption key under question and the public keys of the parties. In my mind it would sound logical that $$C_a' == C_b$$ and this would be a nice check that the re-encryption key is valid, but I could not find any information regarding such checks. Actually, I could not find any public checks that verify a re-encryption key is valid.

Can $$C'_a == C_b$$ ?