hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
A function spectrum or mapping spectrum is the analog of a mapping space in the context of stable homotopy theory. It makes the stable homotopy category into a closed category, and together with the smash product of spectra into a smmyetric closed monoidal category
Given two spectra $X$ and $E$, their function spectrum $F(X,E)$ is the internal hom in a suitable category of spectra.
In the context of generalized (Eilenberg-Steenrod) cohomology the generalized $E$-cohomology of a topological space $X$ is given by the homotopy groups of the mapping spectrum $[\Sigma_\infty X, E]$.
For symmetric spectra: