Adjoints of projections from comma categories

Question

Let $1$ be the terminal category. Consider functors $1 \overset{C}{\rightarrow} \mathcal{C} \overset{F}{\leftarrow} \mathcal{D}.$ I would like to know if there are simple conditions (like presence/preservation of (co)limits etc.) which could guarantee that the projection functor $(C \downarrow F) \overset{P_{\mathcal{D}}}{\rightarrow} \mathcal{D}$ has left or right adjoints. Here $(C \downarrow F)$ is a comma category.

I am not necessarily asking for a characterization of when such adjoints exist. What I would prefer would be simple sufficient conditions, and descriptions of the resulting adjoints. I am especially interested in the following cases:

(1) Where $F$ is the identity functor (so we have a projection from the coslice category).

(2) The case where $C$ picks out the singleton set in $\mathcal{C}=\textbf{Set}$ (so we have a projection from the category of elements of $F$).

Answer 1

First, let us consider the case of the projection from the coslice category. The projection ${}^{C /} \mathcal{C} \to \mathcal{C}$ preserves initial objects if and only if $C$ is initial. Thus, ${}^{C /} \mathcal{C} \to \mathcal{C}$ has a right adjoint if and only if it is an equivalence of categories, which suggests that it will not be very fruitful to look for right adjoints when answering the general question.

So when does the projection $(C \downarrow F) \to \mathcal{D}$ have a left adjoint?

Proposition. If the comma category $(C \downarrow F)$ has an initial object $(I, i)$ and, for every object $D$ in $\mathcal{D}$, the coproduct $I + D$ exists in $\mathcal{D}$, then the projection $(C \downarrow F) \to \mathcal{D}$ has a left adjoint.

Notice that if $\mathcal{D}$ has an initial object and $(C \downarrow F) \to \mathcal{D}$ has a left adjoint then $(C \downarrow F)$ must also have an initial object. Thus, the proposition is optimal (in the sense of giving a necessary and sufficient condition) if $\mathcal{D}$ has an initial object. However, the proposition also applies when $\mathcal{D}$ does not have an initial object.

The proof consists of two straightforward lemmas:

Lemma 1. The following are equivalent:

• The comma category $(C \downarrow F)$ has an initial object.
• The functor $\mathcal{C} (C, F -) : \mathcal{D} \to \textbf{Set}$ is representable.
• There is a commutative diagram of functors of the form below, $$\require{AMScd} \begin{CD} (C \downarrow F) @>{\cong}>> {}^{I /} \mathcal{D} \\ @VVV @VVV \\ \mathcal{D} @= \mathcal{D} \end{CD}$$ where the horizontal arrows are isomorphisms and the vertical arrows are the evident projections.

Lemma 2. Let $I$ be an object in $\mathcal{D}$. The following are equivalent:

• For every object $D$ in $\mathcal{D}$, the coproduct $I + D$ exists in $\mathcal{D}$.
• The projection ${}^{I /} \mathcal{D} \to \mathcal{D}$ has a left adjoint.