Let $C$ be a locally presentable category. Is it true that the category of pointed objects $C^{\*/}$ in $C$ is also locally presentable?



Pointed objects in a Presentable Category
Yes. The category of pointed objects is the category of algebras of an accessible monad on $C$, namely the monad $c \mapsto \ast \sqcup c$ (that is induced by the unique monoid structure on $\ast$ with respect to the monoidal product $\sqcup$), and such a category of algebras is again locally presentable. See Adámek-Rosický, Locally Presentable and Accessible Categories, 2.78. 


