Direct Image of a Subset

The direct image of a subset U ⊆ X 𝑈 𝑋 U\subseteq X along a function f : X → Y : 𝑓 → 𝑋 𝑌 f:X\to Y , denoted f ∗ ⁢ ( U ) ⊆ Y subscript 𝑓 𝑈 𝑌 f_{*}(U)\subseteq Y , is the set { y : Y } conditional-set 𝑦 𝑌 \left\{\;y:Y\right\} .

When viewed fibrationally, direct images arise as Cocartesian Lifts in the Fibration of Subsets.