compact object as filtered colimit and retract

Let \(\mathcal{C}\) be an infinity category, \(\kappa\) a regular cardinal and \(c \in \mathcal{C}\) a kappa compact object.
Assume that \(c\) is a kappa filtered infinity colimit

\begin{align*} c = \mathrm{colim} s_i \end{align*}

for some \(s_i\)

Then \(c\) is a infinity retract of some \(s_j\)

apply the infinity homotopy category functor preserves filtered colimits, which reduces this to the \(1\)-categorical case

\begin{align*} \mathrm{hocat}_{\mathcal{C}}(c) =& \mathrm{hocat}_{\mathcal{C}}(\mathrm{colim} s_i) \\ \cong& \mathrm{colim} \mathrm{hocat}_{\mathcal{C}}(s_i) \end{align*}

Hence we may asusme that \(\mathcal{C} = \mathrm{hocat}(\mathcal{C})\)
Then as \(c\) is \(\kappa\)-compact, we get the isomorphism

\begin{align*} \mathrm{Hom}_{\mathcal{C}}(c, \mathrm{colim}(s_i)) \cong& \mathrm{colim} \mathrm{Hom}_{\mathcal{C}}(c, s_i) \end{align*}

Then tracing the identity, we may find an element \(f: c \rightarrow s_i\) in some \(\mathrm{Hom}_{\mathcal{C}}(c,s_i)\).
This gives a retraction (todo)