Here $\mathcal{C}$ is a presentable infinity category. In the proof of proposition 1.4.3.4 in higher algebra, the fact that $\Omega^\infty : \operatorname{Sp}(\mathcal{C})\rightarrow \mathcal{C}$ is accessible is used. I'm not sure why this is true.
I think this is true because: $\operatorname{Sp}(\mathcal{C})=\operatorname{Exc}_*(\mathcal{S}_*^{\text{fin}},\mathcal{C})$ and $\Omega^\infty$ is given by evaluating at $S^0$, if colimits are computed pointwise in $\operatorname{Exc}_*(\mathcal{S}_*^{\text{fin}},\mathcal{C})$ then this should be enough. However I should admit I am still a little confused on the notion of an accessible functor. A functor is accessible if it $\kappa$-accessible for some regular cardinal $\kappa$. But how would we show that there is some $\kappa$ such that $\Omega^\infty$ is $\kappa$-accessible?