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Definition

A locally full sub 2-category is one whose embedding 2-functor $C\to D$ is locally fully faithful. This means that each $C(c_1,c_2)$ is a full subcategory of $D(c_1,c_2)$.

Examples

• The sub-2-category of ${\mathrm{Prof}}_{\mathrm{rep}}\hookrightarrow$ Prof on all representable profunctors is locally full.

• The sub-2-category of the 2-category $T{\mathrm{Alg}}_l$ of algebras for a 2-monad and lax morphisms between them contains, as a locally full sub-2-category, the 2-category $T{\mathrm{Alg}}_p$ of algebras and pseudo morphisms (or of strict morphisms, if $T$ is strict).

Both of these examples are also wide subcategories. A wide and locally full sub-2-category is equivalent to an F-category. See also 2-category equipped with proarrows.