Natural transformations of functors $\rm Cat\to Cat$

Question

For any two categories $C$, $D$ denote by $[C,D]$ the category of functors $C\to D$. Fixed two categories $C$, $C'$, there are two functors $\rm Cat\to Cat$, namely $[C,-]$ and $[C',-]$; suppose to have a natural transformation $\varphi:[C,-]\Rightarrow [C',-]$ induced by a functor $f:C'\to C$.

Fixed any category $D$, the map $\operatorname{mor} \varphi_D:\operatorname{mor}[C,D]\to \operatorname{mor}[C',D]$ is "equal" to the map $\operatorname{ob} \varphi_{[2,D]}:\operatorname{ob}[C,[2,D]]\to \operatorname{ob}[C',[2,D]]$.

Suppose now that $\varphi$ factors through a subfunctor $S\subseteq[C',-]$, for some natural transformation $\phi:[C,-]\Rightarrow S$. It can happen, for any $D$, that:

1. $\operatorname{ob} S([2,D])=\operatorname{mor}S(D) $;
2. also $\operatorname{ob} \phi_{[2,D]}=\operatorname{mor} \phi_D$.

For example when $C'$ is a model category, $C:=\operatorname {Ho} C'$, $f$ is the canonical functor and $S:=[C',-]_h$ (the full subcategory of $[C',-]$ containing only the functors sending weak equivalences to isomorphisms).

Does the hypothesis that $\varphi$ factors through a subfunctor $S\subseteq[C',-]$, for a natural transformation $\phi:[C,-]\Rightarrow S$, imply automatically that $S$ and $\phi$ satisfy the properties 1 and 2 above? I really think not, but I have troubles to come up with a counterexample. Thank you in advance.

Answer 1

Let me describe the abstract situation first.

In a 2-category $\mathcal{K}$, given an object $K$ and a category $C$, the power (or cotensor) $K^C$ is an object equipped with isomorphisms $$\mathcal{K} (K', K^C) \cong [C, \mathcal{K} (K', K)]$$ that are 2-natural in $K'$. Slightly more explicitly, that means we have a functor $\pi : C \to \mathcal{K} (K^C, K)$ such that, for every object $K'$ in $\mathcal{K}$, the functor $\mathcal{K} (K', K^C) \to [C, \mathcal{K} (K', K)]$ defined by $f \mapsto (c \mapsto \pi (c) \circ f)$ is an isomorphism of categories.

Given a 2-functor $F : \mathcal{K} \to \mathcal{L}$, we get a canonical morphism $F (K^C) \to F (K)^C$ whenever these powers exist: $$\require{AMScd} \begin{CD} \mathcal{K} \left( K^C, K^C \right) @>{\cong}>> \left[ C, \mathcal{K} (K^C, K) \right] \\ & @VV{[C, F]}V \\ \mathcal{L} \left( F (K^C), F (K)^C \right) @>>{\cong}> \left[ C, \mathcal{L} \left( F (K^C), F (K) \right) \right] \end{CD}$$ (Chase $\textrm{id}_{K^C}$ in the above diagram!) We say $F$ preserves powers if $F (K^C) \to F (K)^C$ is an isomorphism for all $K$ and $C$.

For example, $\textbf{Cat}$ has powers: $D^C$ can be identified with $[C, D]$. Thus, for any 2-category $\mathcal{K}$, representable 2-functors $\mathcal{K} \to \textbf{Cat}$ preserve powers.

You are basically asking:

1. Given a 2-functor $G : \mathcal{K} \to \mathcal{L}$ that preserves powers and a sub-2-functor $G' \subseteq G$, does $G'$ also preserve powers?

2. Assuming $\mathcal{K}$ and $\mathcal{L}$ have powers by $\mathbf{2}$ and $\mathbf{3}$, given 2-functors $F, G : \mathcal{K} \to \textbf{Cat}$ that preserve those powers, does every natural transformation $\textrm{ob} \circ F \Rightarrow \textrm{ob} \circ G$ (of functors $\mathcal{K} \to \textbf{Set}$) come from a 2-natural transformation $F \Rightarrow G$?

The answer to 1 is, of course, no. For example, we could take $\mathcal{L} = \textbf{Cat}$, $G$ any representable 2-functor, and $G' = \emptyset$: assuming $\mathcal{K}$ is not empty, then $G' (K^\emptyset) \not\cong G (K)^\emptyset$.

The answer to 2 is yes. As you say, the trick is to identify $\operatorname{mor} F (K)$ with $\operatorname{ob} F (K^\mathbf{2})$. This identification has two parts: the first is identifying $\operatorname{mor} F (K)$ with $\operatorname{ob} {[\mathbf{2}, F (K)]}$, which is as canonical as could possibly be; the second is identifying $[\mathbf{2}, F (K)]$ with $F (K^\mathbf{2})$, which is canonical contingent on the 2-functoriality of $F$. The unique functor $\mathbf{2} \to \mathbf{1}$ corresponds to the map $\operatorname{ob} F (K) \to \operatorname{mor} F (K)$ sending each object to its identity morphism, which helps us establish that the map $\operatorname{mor} F (K) \to \operatorname{mor} G (K)$ preserves identities. The same trick with $\mathbf{3}$ instead of $\mathbf{2}$ helps establish that the induced map $\operatorname{mor} F (K) \to \operatorname{mor} G (K)$ preserves composition. So voilà, we have extended a natural map $\operatorname{ob} F (K) \to \operatorname{ob} G (K)$ to a natural functor $F (K) \to G (K)$, and it is even 2-natural by construction.

(Since we are assuming that $F$ and $G$ preserve powers, a natural functor $F (K) \to G (K)$ is 2-natural if and only if the composite $F (K^\mathbf{2}) \to G (K^\mathbf{2}) \cong G (K)^\mathbf{2}$ is equal to the composite $F (K^\mathbf{2}) \cong F (K)^\mathbf{2} \to G (K)^\mathbf{2}$, but this is automatic because we used exactly this to define the action of $F (K) \to G (K)$ on morphisms in the first place!)