On p.60 of Moritz Groth short course on $\infty$-category he writes, If $C$ is an $\infty$-category adding a disjoint base point defines a functor $$_+:C \rightarrow C_{*/}$$ where $C_{*/}$ is the slice category over $*$. How is this defined?
Thoughts: I know a morphism $* : \Delta^0 \rightarrow C$ determines a distinct point on each level $C_n$. By adjunction, defining a map $C \rightarrow C_*$ is equivalent to a morphism $$\Delta^0 \star C \rightarrow C $$ The natural map seems to be collapse on each level $n$ everything else other than $C_n$, which we take to be identity.
In HTT, 7.2.2.1 the notion of pointed objects is defined. For the equivalence with how Moritz Groth defined this, see HTT, 7.2.2.8. As for your question, if $C$ has finite coproducts then the functor you want is simply the cobase change along $\emptyset \to \ast$. Formally the functor can be defined by using the theory of Cartesian and Cocartesian fibrations, see HTT, 6.1.1 for a discussion of the dual notion of base change and apply this to $C^{\mathrm{op}}$ to obtain the result that you are after.