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The animal was discovered by a member of the public outside an unstaffed sanctuary, meaning it would have been trapped in the box all night.
The Scottish SPCA are now appealing for witnesses to help trace the owner of the cat after the discovery, outside on the cat shelter on Casselbank Street on 17 September at around 5.30pm.
Due to the time of the evening, the welfare organisation building was not staffed.
CCTV shows a woman leaving the cat carrier at the address and ringing the bell.
The SSPCA is also concerned that the cat could potentially have been mistaken for a stray before being brought in and are looking to get in touch with the potential owner.
Scottish SPCA inspector Fiona Thorburn said: “The cat is a young female and black in colour. She’s in good body condition and doesn’t appear to have any health issues.
“We are very grateful to the passer-by who spotted the cat and called us. The cat would have been out there all night on her own which would have been very stressful and who knows what may have happened.
“Unfortunately she is not chipped so we haven’t been able to trace an owner.
“We have no way of knowing whether the woman caught on CCTV is the owner or someone who may have found the cat on the street and assumed she was a stray. We are concerned someone might be looking for her.
“But if the cat was intentionally abandoned at the Cats Protection League then we do understand the owner may have meant for her to be found but this is not the way to go about giving up your pet if you can no longer care for them.
“In these situations we are always keen to speak to the person so that we can find out the circumstances and so that we are aware of an animal’s veterinary history.
“If anyone has any information on this incident they can contact our confidential animal helpline on 03000 999 999.”