Dick knows how lonely a childhood at Wayne's Manor can be and that is why he visits so much, so at least Damian could pretend to be bothered about being asked how he is doing at school, but talking in 10 minutes more than he talks with Bruce one week.
So he is grateful, that at least Bruce was thoughtful enough to get the kid a dog. He probably remembered a young 11 year old and how a dog call Ace make his life happier.
Dick therefore, thinks Titus is a great addition to the family. Besides being a energetic stubborn dog, rough enough for his rough little brother, keeps the boy company and Damian being the kid he is, tells the dog about things he would never discuss with adults. He has listen those conversations (sue me, once a Robin...)He talks about the warm mornings of his childhood, school, some movement he has mastered , and sometimes about his arguments with Bruce. He never intervenes because once he did the same and understands the peace that brings sometimes to let everything out.
Also seems Titus is a great injuries alarm. Bruce was telling him about a mission the youngest bird got hurt but hide it. When they came home Titus received them an start sniffing Damian and whining. Damian tried to stop it, but the dog being more stubborn than the kid (imagine that) was no distracted. After a few minutes and an Alfred-glare, the little bat confess his crime and got some medical attention.
Dick smiles at the scene unfolding in front of him. Titus also keeps Damian warm.
The kid is sleeping on his bed curled to the big dog, who is comfortably lying under the blankets, proof enough Damian let it on bed. Even if Damian denies it, he finds Gotham being too cold for him, so without doubt the warm body next to him, is a nice addition.
The dog is staring at Dick, not moving but also on guard (Titus loves Dick...he makes the most amazing belly rubs and gives him beacon). Dick knows that when he leaves the dog will continue on guard protecting its young ward until he is sure there is no threat.
Yep, he definitely likes Titus. Now he just has to make Damian stop training him to pee on Tim's stuff and eating his shoes.