Adjunction for infinity category of pointed objects

Question

Let C be an infinity category, which admits finite colimits. In Higher Algebra Lemma 1.4.2.19. it is claimed that the forgetful functor $f:\mathcal{C}_{*} \rightarrow \mathcal{C}$ from the infinity category of pointed objects in $\mathcal{C}$ to $\mathcal{C}$ admits a left adjoint.
My idea was to construct the left adjoint as the the functor given on vertices as $C \mapsto (1 \rightarrow C\coprod 1)$, where 1 is a final object of $\mathcal{C}$, but I was not able to show that this functor is indeed a left adjoint. It would be great, if someone could show me how to prove this.

Answer 1

Your idea works indeed and is in fact already enough to construct a left adjoint as required. In general, constructing a functor of $\infty$-categories requires a lot more than just defining it on 0-simplices. However, amazingly it does suffice to construct left adjoint functors 0-simplex-wise. The following lemma makes this precise.

Lemma (Higher Topos Theory, Proposition 5.2.2.12). Let $f\colon \mathcal C\rightarrow \mathcal D$ be a functor of $\infty$-categories. Suppose that the induced functor $\mathrm{h}f\colon \mathrm{h}\mathcal C\rightarrow \mathrm{h}\mathcal D$, enriched in the homotopy category $\mathcal H$ of spaces (or "anima"), admits a left/right adjoint. Then $f$ admits a left/right adjoint (which is unique up to contractible choice and given by a left/right adjoint of $\mathrm{h}f$ on 0-simplices).

So it suffices to show that $c\mapsto (1\rightarrow c\sqcup 1)$ constitues an $\mathcal H$-enriched left adjoint of $\mathrm{h}f\colon \mathrm{h}\mathcal C_*\rightarrow\mathrm{h}\mathcal C$. Let $1'\rightarrow c'$ be another object of $\mathcal C_*$. We claim that there is a canonical equivalence $$\operatorname{Map}_{\mathcal C_*}((1\rightarrow c\sqcup 1),(1'\rightarrow c'))\simeq \operatorname{Map}_{\mathcal C}(1,1')\times_{\operatorname{Map}_{\mathcal C}(1,c')}^{\mathbf R}\operatorname{Map}_{\mathcal C}(c\sqcup 1,c')\,.\tag{$1$}$$ Here $-\times_-^{\mathbf R}-$ denotes the homotopy pullback in $\mathcal H$. Assuming $(1)$ for the moment, observe that $\operatorname{Map}_{\mathcal C}(1,1')\simeq *$ because $1'$ is final, and $\operatorname{Map}_{\mathcal C}(c\sqcup 1,c')\simeq \operatorname{Map}_{\mathcal C}(c,c')\times \operatorname{Map}_{\mathcal C}(1,c')$ by definition, hence $(1)$ provides a functorial equivalence $$\operatorname{Map}_{\mathcal C_*}((1\rightarrow c\sqcup 1),(1'\rightarrow c'))\simeq \operatorname{Map}_{\mathcal C}(c,c')\,,\tag{$2$}$$ which proves that $c\mapsto (1\rightarrow c\sqcup 1)$ is indeed an $\mathcal H$-enriched left adjoint of $\mathrm{h}f$

To prove $(1)$, recall that $\mathcal C_*$ is a full $\infty$-subcategory of the arrow category $\operatorname{Arr}(\mathcal C)=\operatorname{Fun}(\Delta^1,\mathcal C)$ and that, in general, mapping spaces in arrow categories can be computed by a homotopy pullback as in $(1)$. I don't know if there is a canonical reference for this fact (and I'd be happy if someone could suggest one), but I learned it from Fabian Hebestreits lectures. See Proposition VIII.5 in these lecture notes (be aware that Fabian uses "$\operatorname{Hom}_{\mathcal C}$" instead of "$\operatorname{Map}_{\mathcal C}$", and "anima" instead of "Kan complex/space/$\infty$-groupoid").