Full subcategory inclusions

Definition 1.5.9 (Full subcategory). A functor \(i\colon A \to C\) is called a full subcategory inclusion if \(i^{\simeq } \colon A^{\simeq } \to C^{\simeq }\) is a monomorphism, and for every \(\infty \)-category \(T\) the commutative square

is a pullback square. The vertical functors are constructed by restricting evaluation to \(\Map (T,A)\times T^{\simeq }\) and \(\Map (T,C)\times T^{\simeq }\), factoring through the corresponding groupoid cores, and currying.