nLab enriched Yoneda lemma




The enriched Yoneda lemma is the generalization of the usual Yoneda lemma from category theory to enriched category theory.


We discuss here two forms of the Yoneda lemma.

Let VV be a (locally small) closed symmetric monoidal category, so that VV is enriched in itself via its internal hom.

Weak form

A weak form of the enriched Yoneda lemma says that given a VV-enriched functor F:CVF: C \to V and an object cc of CC, the set of VV-enriched natural transformations α:hom C(c,)F\alpha: \hom_C(c, -) \to F is in natural bijection with the set of elements of F(c)F(c), i.e., the set of morphisms IF(c)I \to F(c), obtained by composition:

I1 chom C(c,c)αcF(c)I \stackrel{1_c}{\to} \hom_C(c, c) \stackrel{\alpha c}{\to} F(c)

Strong form

Now suppose that VV is in addition (small-)complete (has all small limits). Then, given a small VV-enriched category CC and VV-enriched functors F,G:CVF, G: C \to V, one may construct the object of VV-natural transformations as an enriched end:

V C(F,G)= cV(F(c),G(c))V^C(F, G) = \int_c V(F(c), G(c))

(which may in turn be expressed as an ordinary limit in VV). This is the hom-object in the enriched functor category.

A strong form of the enriched Yoneda lemma specifies a VV-natural isomorphism

V C(hom C(d,),F)= cV(hom C(d,c),F(c))F(d).V^C(\hom_C(d, -), F) = \int_c V(\hom_C(d,c),F(c)) \cong F(d).

This implies the weak form by applying the functor hom(I,):VSet\hom(I, -): V \to Set.

Density form

If VV is also (small-)cocomplete, there is an equivalent dual formulation of the enriched Yondeda lemma as a VV-natural isomorphism

chom C(c,d)F(c)F(d)\int^c \hom_C(c,d) \otimes F(c) \cong F(d)

induced by the morphisms

hom C(c,d)F(c)F(d)\hom_C(c,d) \otimes F(c) \to F(d)

adjoint to F c,dF_{c,d}. This version is sometimes called the density theorem and sometimes simply called the enriched Yoneda lemma.

Dual forms

For an enriched presheaf H:C opVH: C^{op} \to V there are dual forms obtained by replacing CC above with C opC^{op}. These are

cV(hom C(c,d),H(c))H(d) \int_c V(\hom_C(c,d),H(c)) \cong H(d)


chom C(d,c)H(c)H(d). \int^c \hom_C(d,c) \otimes H(c) \cong H(d).


  • Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 (pdf)

Textbook accounts include

Generalizations to the case that the enriching monoidal category is not closed or symmetric (using skew-symmetric categories? or tensored categories?) can be found in

Last revised on May 3, 2023 at 10:48:35. See the history of this page for a list of all contributions to it.