Definition of "functor"

Choosing for U {\displaystyle U} the operation of closure, regularization or relative completion, we obtain from a given functor F ∈ J F {\displaystyle {\mathcal {F}}\in {\mathcal {JF}}} the functors F ¯ : X → → F ( X → ) ¯ , F 0 : X → → F ( X → ) 0 , F c : X → → F ( X → ) c {\displaystyle {\overline {F}}:{\overrightarrow {X}}\rightarrow {\overline {F({\overrightarrow {X}})}},F^{0}:{\overrightarrow {X}}\rightarrow F({\overrightarrow {X}})^{0},F^{c}:{\overrightarrow {X}}\rightarrow F({\overrightarrow {X}})^{c}} .

1991, Natalie Wadhwa (translator), Yu. A. Brudnyǐ, N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces, Volume I, Elsevier (North-Holland), page 143

Given a homotopical category X {\displaystyle X} and a functor u : A → B {\displaystyle u:A\rightarrow B} , a homotopical u {\displaystyle u} -colimit (resp. u {\displaystyle u} -limit) functor on X {\displaystyle X} will be a homotopically terminal (resp. initial) Kan extension of the identity (50.2) along the induced diagram functor X u : X B → X A {\displaystyle X^{u}:X^{B}\rightarrow X^{A}} (47.1).

2004, William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Jeffrey H. Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories, American Mathematical Society, page 165

In this chapter, we recall the definition of the category of Σ ∗ {\displaystyle \Sigma _{*}} -objects and we review the relationship between Σ ∗ {\displaystyle \Sigma _{*}} -objects and functors. In short, a Σ ∗ {\displaystyle \Sigma _{*}} -object (in English words, a symmetric sequence of objects, or simply a symmetric object) is the coefficient sequence of a generalized symmetric functor S ( M ) : X → S ( M , X ) {\displaystyle S(M):X\rightarrow S(M,X)} , defined by a formula of the form S ( M , X ) = ⨁ r = 0 ∞ ( M ( r ) ⊗ X ⊗ r ) Σ r {\displaystyle S(M,X)=\bigoplus _{r=0}^{\infty }\left(M(r)\otimes X^{\otimes r}\right)_{\Sigma _{r}}} .

2009, Benoit Fresse, Modules Over Operads and Functors, Springer, Lecture Notes in Mathematics: 1967, page 35