In Remark 5.4.2.15 in Higher Topos Theory, Lurie explains in what sense an accessible functor $F:\mathcal{C}\rightarrow \mathcal{D}$ between accessible $\infty$-categories is "determined by small data". In particular he wants to show that $F$ is induced by a functor $F':\mathcal{C}^\kappa\rightarrow \mathcal{D}^\kappa$, where $\mathcal{C}^\kappa$ refers to the $\kappa$-compact objects of $\mathcal{C}$.


He argues as follows. We begin by picking a regular cardinal $\tau$ such that $\mathcal{C}$, $\mathcal{D}$ and $F$ are $\tau$-accessible. Now $\mathcal{C}\simeq \mathrm{Ind}\_\tau(\mathcal{C}^\tau)$, and so the universal property tells us that $F$ is induced by a functor $F':C^\tau\rightarrow \mathcal{D}$.


This is where I lose the plot. Lurie then seems to claim that remark 5.4.2.13 implies that the image of $F'$ is contained in the $\tau'$-compact objects of $\mathcal{D}$ for some other $\tau'$. However all that remark 5.4.2.13 says is that given any regular cardinal $\tau'$, $\mathcal{D}^{\tau'}$ is small. We also know that the image of $F'$ is small, but nevertheless I don't see why this is enough to conclude what Lurie requires. So the question is the following: Why does the image of $F'$ land inside the $\tau'$-compact objects of $\mathcal{D}$ for some regular cardinal $\tau'$?


Ps. After concluding that the image of $F'$ lands in $\mathcal{D}^{\tau'}$, Lurie claims that we can pick $\kappa>> \tau'$ and obtain our conclusion. But again I fail to understand how we still know that $F(\mathcal{C}^\kappa)\subset \mathcal{D}^\kappa$.



Remark 5.4.2.15 in HTT
He is applying 5.4.2.13 to $C$ and not to $D$: Because $C^\kappa$ is essentially small, and each $F(c)$ is $\lambda\_c$-compact for some $\lambda$; there is a $\kappa = \sup\_{c \in C^\kappa} \lambda\_c$ such that each $F(c)$ is $\kappa'$-compact (well, rather, take some regular cardinal bigger than $\sup\_{c \in C^\kappa} \lambda\_c$, and by $c \in C^\kappa$, I really, mean for $c$ running in a small set of object representing all isomorphisms class of objects in $C^\kappa$.


