Yoneda Lemma when there is a monad

[info@christophertownsend.org]

Hi

Does anyone have any references for the following generalisation of
the Yoneda lemma:

Lemma: If (T,i,m) is a monad on a locally small category C then for
any functor F:(C_T)^op->Set, contravariant from the Kleisli category
to Set and for any T algebra (A,a)

Nat[C^T(K_,(A,a)),F]=F(A,a)

where K is the usual comparison functor from the Kleisli category,
C_T, to the category of algebras of T, C^T.  F(A,a) means the subset
of F(A) consisting of elements x such that F(m_A)x=F(Ta)x. (And Nat[_]
means the set of natural transformations.) //

The usual Yoneda lemma is recovered by taking the trivial monad. The
Lemma gives a generalised Yoneda embedding: C^T embeds in
[(C_T)^op,Set]; i.e. any category of algebras embeds in the presheaf
category of the Kleisli category. I wasn't aware of this quite trivial
result and was hoping for some guidance as to where it is covered
already in the literature?

Thanks, Christopher




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