equivalences in/of $(\infty,1)$-categories
The notion of locally small $(\infty,1)$-category is the generalization of the notion of locally small category from category theory to (∞,1)-category theory.
A quasi-category $C$ is locally small if for all objects $x,y \in C$ the hom ∞-groupoid $Hom_C(x,y)$ is essentially small.
This appears as HTT, below prop. 5.4.1.7.
A quasi-category $C$ is locally small precisely if the following equivalent condition holds:
for every small set $S$ of objects in $C$, the full sub-quasi-category on $S$ is essentially small.
This is the topic of section 5.4.1 of
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