I've seen the following theorem around in various forms:


To give an object $A \in \mathcal{C}$ the structure of a $\Omega$-algebra object in $\mathcal{C}$ is equivalent to giving a lift of the contravariant hom-functor $\mathcal{C}(-,A) \colon \mathcal{C} \to Set$ to a functor $\mathcal{C} \to \Omega$.


Here $\Omega$ denotes the category of $\Omega$-algebra objects in $Set$.


However, despite much searching I've been unable to find a proof in the reverse direction. My particular question is given a lift of the hom-functor, how can we define the maps giving $A$ the structure of a $\Omega$-algebra object.


EDIT: After the answer by Dimitri, here's my attempt at a proof:


Let $h\_A = \mathcal{C}(-,A) \colon \mathcal{C}\to Set$ and suppose that this lifts.


$h\_A$ is an $\Omega$-algebra object in the functor category $[\mathcal{C}, Set]$. Let's suppose for an example that our $\Omega$-algebra, has a multiplication $\mu$. Then we have a natural transformation $\mu \colon h\_A \times h\_A \to h\_A$.


The Yoneda embedding now tells us we have an isomorphism $Nat(h\_{A \times A}, h\_A) = \mathcal{C}(A \times A, A)$. Since $h\_{A \times A}$ is isomorphic to $h\_A \times h\_A$ we see our natural transformation specifies a map $A \times A \to A$. 


Thanks


Will



Algebraic objects and lifts of their represented functors
Giving a functor $C \rightarrow \Omega$ is the same as giving an object with an $\Omega$-algebra structure in the category of functors $[C, Set]$. Having observed this the theorem follows form Yoneda and the fact that it preserves products.


