Part of me will always hate the headcanons that imply Alfred the Cat/Titus/Batcow are anyone's pets but Damian's. Every single one of them have a very specific backstory that is very important to Damian's character and relationships; they aren't the family pets. It's like taking Ace from Bruce. That's his fucking dog, man.
Alfred the Cat was given to Damian by Alfred when he was fresh to the manor. Alfred went to the shelter, saw a kitten that was lashing out out of fear and needed a little affection to open up. Alfred gave this cat to Damian specifically so he could see himself in it, be kind to himself, prove to himself that he does have good within him. Damian trained that cat, was so patient with that cat, there are specific panels showing his arms scratched up from his persistence. He named the cat Alfred, not to be cute and cheeky, but because Alfred was one of the first people to show him that sort of grace.
Batcow isn't just a funny haha everything is bat themed, even the cow. Damian and Dick rescued her from being experimented on, she was in a slaughter house with gore everywhere. Damian's standing in a puddle of animal blood, post-throwing up from how disgusting it was, declaring that he's going to be a vegetarian. The juxtaposition of that to the kid who was an assassin only months ago in universe? Come on.
Titus was given to Damian by Bruce specifically because he was having a hard time bonding with him. Damian rejected the dog, refused to name him and called him a nuisance. After Bruce expresses his reasoning, that he's proud of Damian and his growth, his restraint, and the dog is just a gift, only then does Damian show more acceptance for him. For both Titus and Bruce. Titus is a one-to-one reflection of Damian's relationship with his father, he isn’t the "family dog."
