nLab final functor




A functor F:CDF \colon C \to D is final, if restricting diagrams along FF does not change their colimit.

Dually, a functor is initial if pulling back diagrams along it does not change the limits of these diagrams.

Beware that this property is pretty much unrelated to that of a functor being an initial object or terminal object in the functor category [C,D][C,D]. The terminology comes instead from the fact that an object dDd\in D is initial (resp. terminal) just when the corresponding functor d:1Dd \colon 1\to D is initial (resp. final).


(warning on terminology)
In older references (and also some others like HTT), final functors are sometimes called cofinal, the terminology having been imported from order theory (cf. cofinality). However, this is confusing in category theory because usually the prefix “co-” denotes dualization. In at least one place (Borceux) this non-dualization was treated as a dualization and the word “final” used for the dual concept, but in general it seems that the consensus is to use “final” for what used to be called “cofinal”, and “initial” for the dual concept (since “co-final” would be ambiguous). For example, Johnstone in Sketches of an Elephant says (before Proposition B2.5.12 ):

Traditionally, final functors were called ‘cofinal functors’; but this use of ‘co’ is potentially misleading as it has nothing to do with dualization — it is derived from the Latin ‘cum’ rather than ‘contra’ — and so it is now generally omitted.

See also the warning at final ( , 1 ) (\infty,1) -functor (here).



A functor F:CDF : C \to D is final if for every object dDd \in D the comma category (d/F)(d/F) is (non-empty and) connected (the non-emptiness condition is redundant since connected categories are non-empty by convention).

A functor F:CDF : C \to D is initial if the opposite F op:C opD opF^{op} : C^{op} \to D^{op} is final, i.e. if for every object dDd \in D the comma category (F/d)(F/d) is connected.



Let F:CDF : C \to D be a functor

The following conditions are equivalent.

  1. FF is final.

  2. For all functors G:DSetG : D \to Set the natural function between colimits

    lim GFlim G \lim_\to G \circ F \to \lim_{\to} G

    is a bijection.

  3. For all categories EE and all functors G:DEG : D \to E the natural morphism between colimits

    lim GFlim G \lim_\to G \circ F \to \lim_{\to} G

    is an isomorphism.

  4. For all functors G:D opSetG : D^{op} \to Set the natural function between limits

    lim Glim GF op \lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}

    is a bijection.

  5. For all categories EE and all functors G:D opEG : D^{op} \to E the natural morphism

    lim Glim GF op \lim_\leftarrow G \to \lim_\leftarrow G \circ F^{op}

    is an isomorphism.

  6. For all dDd \in D

    lim cCHom D(d,F(c))*. {\lim_\to}_{c \in C} Hom_D(d,F(c)) \simeq * \,.
  7. for all functors G:DEG: D \to E, the canonical map

    Nat(G,Δ)Nat(GF,Δ):ESet Nat(G,\Delta-)\to Nat(GF,\Delta-) : E \to Set

    between the functors of cocones on GG and GFGF is an isomorphism. (Here, the Δ\Delta‘s denote the functors sending objects to constant functors, and NatNat stands for the set of natural transformations.)


If F:CDF : C \to D is final then CC is connected precisely if DD is.


If F 1F_1 and F 2F_2 are final, then so is their composite F 1F 2F_1 \circ F_2.

If F 2F_2 and the composite F 1F 2F_1 \circ F_2 are final, then so is F 1F_1.

Final functors are stable under pushout.

The coproduct of final functors in the arrow category Cat 2\mathbf{Cat}^{\mathbf{2}} is a final functor.

If F 1F_1 is a full and faithful functor and the composite is final, then both functors seperately are final.

The first two statements of Proposition in fact follow from the stability properties of orthogonal factorization systems:


Exact squares

The characterization of final functors is a special case of the characterization of exact squares.

Final enriched functors

Finality for enriched functors with respect to weighted colimits is discussed in Kelly 1982 §4.5.

See also discussion of finality specifically for coends (initiality for weighted limits and ends) by:

Final \infty-functors

The generalization of the notion of final functor from category theory to (∞,1)-higher category theory is described at



(inclusion of a terminal object is final functor)
If DD has a terminal object then the functor F:*DF : {*} \to D that picks that terminal object is final: for every dDd \in D the comma category d/Fd/F is equivalent to **. The converse is also true: if a functor *D*\to D is final, then its image is a terminal object.

In this case the statement about preservation of colimits states that the colimit over a category with a terminal object is the value of the diagram at that object. Which is also readily checked directly.

More generally, a functor whose domain is a discrete category is final if and only if it is a right adjoint.


A functor between groupoids is final iff it is essentially surjective and full.

(eg. Cigoli 2018, Prop. 1.1)


Every right adjoint functor is final.


Let (LR):CD(L \dashv R) : C \to D be a pair of adjoint functors.To see that RR is final, we may for instance check that for all dDd \in D the comma category d/Rd / R is non-empty and connected:

It is non-empty because it contains the adjunction unit (L(d),dRL(d))(L(d), d \to R L (d)). Similarly, for

d f g R(a) R(b) \array{ && d \\ & {}^{\mathllap{f}}\swarrow && \searrow^{\mathrlap{g}} \\ R(a) &&&& R(b) }

two objects, they are connected by a zig-zag going through the unit, by the universal factorization property of adjunctions

d R(a) Rf¯ RL(d) R(g¯) R(b). \array{ && d \\ & \swarrow &\downarrow& \searrow \\ R(a) &\stackrel{R \bar f}{\leftarrow}& R L (d)& \stackrel{R(\bar g)}{\to} & R(b) } \,.

The inclusion 𝒞𝒞˜\mathcal{C} \to \tilde \mathcal{C} of any category into its idempotent completion is final.

See at idempotent completion in the section on Finality.


The inclusion of the cospan diagram into its cocone

(a c b)(a c p b) \left( \array{ a \\ \downarrow \\ c \\ \uparrow \\ b } \right) \hookrightarrow \left( \array{ a \\ \downarrow & \searrow \\ c &\longrightarrow & p \\ \uparrow & \nearrow \\ b } \right)

is initial.


By the characterization (here) of limits in a slice category, this implies that fiber products in a slice category are computed as fiber products in the underlying category, or in other words that dependent sum to the point preserves fiber products.


For Δ op\Delta^{op} the opposite of the simplex category, the non-full subcategory inclusion of the lowest two face maps

([1]d 0d 1[0])Δ op \big( [1] \underoverset {d_0} {d_1} {\rightrightarrows} [0] \big) \;\xrightarrow{\;\;\;\;}\; \Delta^{op}

is a final functor.

It follows that the colimit over a simplicial diagram is equivalently the coequalizer of the lowest two face maps.

(e.g. Riehl 14, Exp. 8.3.8)

basic properties of…


In enriched category theory:

  • Max Kelly, §4.5 in: Basic concepts of enriched category theory, London Math. Soc. Lec. Note Series 64, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories 10 (2005) 1-136 [ISBN:9780521287029, tac:tr10, pdf]

In internal category theory

for internal functors between internal groupoids in exact categories:

Last revised on January 18, 2024 at 08:37:12. See the history of this page for a list of all contributions to it.